diff --git a/fsl/data/cifti.py b/fsl/data/cifti.py
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+++ b/fsl/data/cifti.py
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+"""
+Provides a sparse representation of volumetric and/or surface data
+
+The data can be either defined per voxel/vertex (:class:`DenseCifti`) or per parcel (`class:`ParcelCifti`).
+
+The data can be read from NIFTI, GIFTI, or CIFTI files.
+Non-sparse volumetric or surface representations can be extracte.
+"""
+from nibabel.cifti2 import cifti2_axes
+from typing import Sequence
+import numpy as np
+from fsl.data.image import Image
+import nibabel as nib
+from fsl.utils.path import addExt
+
+
+dense_extensions = {
+    cifti2_axes.BrainModelAxis: '.dconn.nii',
+    cifti2_axes.ParcelsAxis: '.dpconn.nii',
+    cifti2_axes.SeriesAxis: '.dtseries.nii',
+    cifti2_axes.ScalarAxis: '.dscalar.nii',
+    cifti2_axes.LabelAxis: '.dlabel.nii',
+}
+
+parcel_extensions = {
+    cifti2_axes.BrainModelAxis: '.pdconn.nii',
+    cifti2_axes.ParcelsAxis: '.pconn.nii',
+    cifti2_axes.SeriesAxis: '.ptseries.nii',
+    cifti2_axes.ScalarAxis: '.pscalar.nii',
+    cifti2_axes.LabelAxis: '.plabel.nii',
+}
+
+
+class Cifti:
+    """
+    Parent class for the two types of CIFTI files.
+
+    The type of the CIFTI file is determined by the last axis, which can be one of:
+
+    - :py:class:`BrainModelAxis <cifti2_axes.BrainModelAxis>`
+    - :py:class:`ParcelsAxis <cifti2_axes.ParcelsAxis>`
+    """
+    def __init__(self, arr: np.ndarray, axes: Sequence[cifti2_axes.Axis]):
+        """
+        Defines a new dataset in greyordinate space
+
+        :param data: (..., N) array for N greyordinates or parcels
+        :param axes: sequence of CIFTI axes describing the data along each dimension
+        """
+        self.arr = arr
+        self.axes = axes
+        if arr.shape[-len(axes):] != tuple(len(ax) for ax in axes):
+            raise ValueError(f"Shape of axes {tuple(len(ax) for ax in axes)} does not match shape of array {self.arr.shape}")
+
+    def to_cifti(self, other_axes=None):
+        """
+        Create a CIFTI image from the data
+
+        :param other_axes: overwrites the :mod:`cifti2_axes` to be used to write to create the CIFTI image
+        :return: nibabel CIFTI image
+        """
+        if other_axes is None:
+            if len(self.axes) != self.data.ndim:
+                raise ValueError("Can not store to CIFTI without defining what is stored along the other dimensions")
+            other_axes = self.axes[:-1]
+        else:
+            if len(other_axes) != self.data.ndim - 1:
+                raise ValueError("Number of axis does not match dimensionality of the data")
+            if tuple(len(ax) for ax in other_axes) != self.data.shape[:-1]:
+                raise ValueError("Size of other axes does not match data size")
+
+        data = self.data
+        if data.ndim == 1:
+            # CIFTI axes are always at least 2D
+            data = data[None, :]
+            other_axes = [cifti2_axes.ScalarAxis(['default'])]
+
+        return cifti2_axes.Cifti2Image(
+            data,
+            header=list(other_axes) + [self.axes[-1]]
+        )
+
+    @classmethod
+    def from_cifti(cls, filename, writable=False):
+        """
+        Creates new greyordinate object from dense CIFTI file
+
+        :param filename: CIFTI filename or :class:`nib.Cifti2Image` object
+        :param writable: if True, opens data array in writable mode
+        """
+        if isinstance(filename, str):
+            img = nib.load(filename)
+        else:
+            img = filename
+
+        if not isinstance(img, nib.Cifti2Image):
+            raise ValueError(f"Input {filename} should be CIFTI filename or nibabel Cifti2Image")
+
+        if writable:
+            data = np.memmap(filename, img.dataobj.dtype, mode='r+',
+                             offset=img.dataobj.offset, shape=img.shape, order='F')
+        else:
+            data = np.asanyarray(img.dataobj)
+
+        axes = [img.header.get_axis(idx) for idx in range(data.ndim)]
+
+        if isinstance(axes[-1], cifti2_axes.BrainModelAxis):
+            return DenseCifti(data, axes)
+        elif isinstance(axes[-1], cifti2_axes.ParcelsAxis):
+            return ParcelCifti(data, axes)
+        raise ValueError("Last axis of CIFTI object should be a BrainModelAxis or ParcelsAxis")
+
+    def write(self, cifti_filename, other_axes=None):
+        """
+        Writes this sparse representation to/from a filename
+
+        :param cifti_filename: output filename
+        :param other_axes: overwrites the :mod:`cifti2_axes` to be used to write to the file
+        :return:
+        """
+        self.to_cifti(other_axes).to_filename(addExt(cifti_filename, defaultExt=self.extension))
+
+    def read(cls, filename, mask_values=(0, np.nan), writable=False):
+        """
+        Reads greyordinate data from the given file
+
+        File can be:
+
+        - NIFTI mask
+        - GIFTI mask
+        - CIFTI file
+
+        :param filename: input filename
+        :param mask_values: which values are outside of the mask for NIFTI or GIFTI input
+        :param writable: allow to write to disk
+        :return: greyordinates object
+        """
+        if isinstance(filename, str):
+            img = nib.load(filename)
+        else:
+            img = filename
+
+        if isinstance(img, nib.Nifti1Image):
+            if writable:
+                raise ValueError("Can not open NIFTI file in writable mode")
+            return cls.from_image(Image(img), mask_values)
+        if isinstance(img, nib.Cifti2Image):
+            return cls.from_cifti(img, writable=writable)
+        if isinstance(img, nib.GiftiImage):
+            if writable:
+                raise ValueError("Can not open GIFTI file in writable mode")
+            return cls.from_gifti(img, mask_values)
+        raise ValueError(f"I do not know how to convert {type(img)} into greyordinates (from {filename})")
+
+    @classmethod
+    def from_gifti(cls, filename, mask_values=(0, np.nan)):
+        """
+        Creates a new greyordinate object from a GIFTI file
+
+        :param filename: GIFTI filename
+        :param mask_values: values to mask out
+        :return: greyordinate object representing the unmasked vertices
+        """
+        if isinstance(filename, str):
+            img = nib.load(filename)
+        else:
+            img = filename
+        datasets = [darr.data for darr in img.darrays]
+        if len(datasets) == 1:
+            data = datasets[0]
+        else:
+            data = np.concatenate(
+                [np.atleast_2d(d) for d in datasets], axis=0
+            )
+        mask = np.ones(data.shape, dtype='bool')
+        for value in mask_values:
+            if value is np.nan:
+                mask &= ~np.isnan(data)
+            else:
+                mask &= ~(data == value)
+        while mask.ndim > 1:
+            mask = mask.any(0)
+
+        anatomy = BrainStructure.from_gifti(img)
+
+        bm_axes = cifti2_axes.BrainModelAxis.from_mask(mask, name=anatomy.cifti)
+        return DenseCifti(data[..., mask], [bm_axes])
+
+    @classmethod
+    def from_image(cls, image, mask_values=(np.nan, 0)):
+        """
+        Creates a new greyordinate object from a NIFTI file
+
+        :param filename: NIFTI filename or Image object
+        :param mask_values: which values to mask out
+        :return: greyordinate object representing the unmasked voxels
+        """
+        img = Image(image)
+
+        mask = np.ones(img.data.shape, dtype='bool')
+        for value in mask_values:
+            if value is np.nan:
+                mask &= ~np.isnan(img.data)
+            else:
+                mask &= ~(img.data == value)
+        while mask.ndim > 3:
+            mask = mask.any(-1)
+
+        inverted_data = np.transpose(img.data[mask], tuple(range(1, img.data.ndim - 2)) + (0, ))
+        bm_axes = cifti2_axes.BrainModelAxis.from_mask(mask, affine=img.affine)
+        return cifti2_axes.GreyOrdinates(inverted_data, [bm_axes])
+
+
+
+class DenseCifti(Cifti):
+    """
+    Represents sparse data defined for a subset of voxels and vertices (i.e., greyordinates)
+    """
+    def __init__(self, *args, **kwargs):
+        super().__init__(self, *args, **kwargs)
+        if not isinstance(self.brain_model_axis, cifti2_axes.BrainModelAxis):
+            raise ValueError(f"DenseCifti expects a BrainModelAxis as last axes object, not {type(self.brain_model_axis)}")
+
+    @property
+    def brain_model_axis(self, ) -> cifti2_axes.BrainModelAxis:
+        return self.axes[-1]
+
+    @property
+    def extension(self, ):
+        return dense_extensions[type(self.axes[-2])]
+
+    def to_image(self, fill=0) -> Image:
+        """
+        Get the volumetric data as an :class:`Image`
+        """
+        if self.brain_model_axis.volume_mask.sum() == 0:
+            raise ValueError(f"Can not create volume without voxels in {self}")
+        data = np.full(self.brain_model_axis.volume_shape + self.data.shape[:-1], fill,
+                       dtype=self.data.dtype)
+        voxels = self.brain_model_axis.voxel[self.brain_model_axis.volume_mask]
+        data[tuple(voxels.T)] = np.transpose(self.data, (-1,) + tuple(range(self.data.ndim - 1)))[
+            self.brain_model_axis.volume_mask]
+        return Image(data, xform=self.brain_model_axis.affine)
+
+    def surface(self, anatomy, fill=np.nan, partial=False):
+        """
+        Gets a specific surface
+
+        If `partial` is True a view of the data rather than a copy is returned.
+
+        :param anatomy: BrainStructure or string like 'CortexLeft' or 'CortexRight'
+        :param fill: which value to fill the array with if not all vertices are defined
+        :param partial: only return the part of the surface defined in the greyordinate file (ignores `fill` if set)
+        :return:
+            - if not partial: (..., n_vertices) array
+            - if partial: tuple with (N, ) int array with indices on the surface included in (..., N) array
+        """
+        if isinstance(anatomy, str):
+            anatomy = BrainStructure.from_string(anatomy, issurface=True)
+        if anatomy.cifti not in self.brain_model_axis.name:
+            raise ValueError(f"No surface data for {anatomy.cifti} found")
+        slc, bm = None, None
+        arr = np.full(self.data.shape[:-1] + (self.brain_model_axis.nvertices[anatomy.cifti],), fill,
+                      dtype=self.data.dtype)
+        for name, slc_try, bm_try in self.brain_model_axis.iter_structures():
+            if name == anatomy.cifti:
+                if partial:
+                    if bm is not None:
+                        raise ValueError(f"Surface {anatomy} does not form a contiguous block")
+                    slc, bm = slc_try, bm_try
+                else:
+                    arr[..., bm_try.vertex] = self.data[..., slc_try]
+        if not partial:
+            return arr
+        else:
+            return bm.vertex, self.data[..., slc]
+
+
+class ParcelCifti(Cifti):
+    """
+    Represents sparse data defined at specific parcels
+    """
+    def __init__(self, *args, **kwargs):
+        super().__init__(self, *args, **kwargs)
+        if not isinstance(self.parcel_axis, cifti2_axes.BrainModelAxis):
+            raise ValueError(f"ParcelCifti expects a ParcelsAxis as last axes object, not {type(self.parcel_axis)}")
+
+    @property
+    def extension(self, ):
+        return parcel_extensions[type(self.axes[-2])]
+
+    @property
+    def parcel_axis(self, ) -> cifti2_axes.ParcelsAxis:
+        return self.axes[-1]
+
+    def to_image(self, fill=0):
+        """
+        Get the volumetric data as an :class:`Image`
+        """
+        data = np.full(self.parcel_axis.volume_shape + self.arr.shape[:-1], fill, dtype=self.arr.dtype)
+        written = np.zeros(self.parcel_axis.volume_shape, dtype='bool')
+        for idx, write_to in enumerate(self.parcel_axis).voxels:
+            if written[write_to].any():
+                raise ValueError("Duplicate voxels in different parcels")
+            data[write_to] = self.arr[np.newaxis, ..., idx]
+            written[write_to] = True
+        if not written.any():
+            raise ValueError("Parcellation does not contain any volumetric data")
+        return Image(data, xform=self.brain_model_axis.affine)
+
+    def surface(self, anatomy, fill=np.nan, partial=False):
+        """
+        Gets a specific surface
+
+        :param anatomy: BrainStructure or string like 'CortexLeft' or 'CortexRight'
+        :param fill: which value to fill the array with if not all vertices are defined
+        :param partial: only return the part of the surface defined in the greyordinate file (ignores `fill` if set)
+        :return:
+            - if not partial: (..., n_vertices) array
+            - if partial: tuple with (N, ) int array with indices on the surface included in (..., N) array
+        """
+        if isinstance(anatomy, str):
+            anatomy = BrainStructure.from_string(anatomy, issurface=True)
+        if anatomy.cifti not in self.parcel_axis.nvertices:
+            raise ValueError(f"No surface data for {anatomy.cifti} found")
+
+        arr = np.full(self.data.shape[:-1] + (self.parcel_axis.nvertices[anatomy.cifti],), fill,
+                      dtype=self.data.dtype)
+        written = np.zeros(self.parcel_axis.nvertices[anatomy.cifti])
+        for idx, vertices in enumerate(self.parcel_axis.vertices):
+            if anatomy.cifti not in vertices:
+                continue
+            write_to = vertices[anatomy.cifti]
+            if written[write_to].any():
+                raise ValueError("Duplicate vertices in different parcels")
+            arr[..., write_to] = self.arr[..., idx, np.newaxis]
+            written[write_to] = True
+
+        if not partial:
+            return arr
+        else:
+            return np.where(written)[0], arr[..., written]
+
+
+class BrainStructure(object):
+    """Which brain structure does the parent object describe?
+
+    Supports how brain structures are stored in both GIFTI and CIFTI files
+    """
+    def __init__(self, primary, secondary=None, hemisphere='both', geometry=None):
+        """Creates a new brain structure
+
+        :param primary: Name of the brain structure (e.g. cortex, thalamus)
+        :param secondary: Further specification of which part of the brain structure is described (e.g. 'white' or
+        'pial' for the cortex)
+        :param hemisphere: which hemisphere is the brain structure in ('left', 'right', or 'both')
+        :param geometry: does the parent object describe the 'volume' or the 'surface'
+        """
+        self.primary = primary.lower()
+        self.secondary = None if secondary is None else secondary.lower()
+        self.hemisphere = hemisphere.lower()
+        if geometry not in (None, 'surface', 'volume'):
+            raise ValueError(f"Invalid value for geometry: {geometry}")
+        self.geometry = geometry
+
+    def __eq__(self, other):
+        """Two brain structures are equal if they could describe the same structure
+        """
+        if isinstance(other, str):
+            other = self.from_string(other)
+        match_primary = (self.primary == other.primary or self.primary == 'all' or other.primary == 'all' or
+                         self.primary == other.geometry or self.geometry == other.primary)
+        match_hemisphere = self.hemisphere == other.hemisphere
+        match_secondary = (self.secondary is None or other.secondary is None or self.secondary == other.secondary)
+        match_geometry = (self.geometry is None or other.geometry is None or self.geometry == other.geometry)
+        return match_primary and match_hemisphere and match_secondary and match_geometry
+
+    @property
+    def gifti(self, ):
+        """Returns the keywords needed to define the surface in the meta information of a GIFTI file
+        """
+        main = self.primary.capitalize() + ('' if self.hemisphere == 'both' else self.hemisphere.capitalize())
+        res = {'AnatomicalStructurePrimary': main}
+        if self.secondary is not None:
+            res['AnatomicalStructureSecondary'] = self.secondary.capitalize()
+        return res
+
+    def __str__(self, ):
+        """Returns a short description of the brain structure
+        """
+        if self.secondary is None:
+            return self.primary.capitalize() + self.hemisphere.capitalize()
+        else:
+            return "%s%s(%s)" % (self.primary.capitalize(), self.hemisphere.capitalize(), self.secondary)
+
+    @property
+    def cifti(self, ):
+        """Returns a description of the brain structure needed to define the surface in a CIFTI file
+        """
+        return 'CIFTI_STRUCTURE_' + self.primary.upper() + ('' if self.hemisphere == 'both' else ('_' + self.hemisphere.upper()))
+
+    @classmethod
+    def from_string(cls, value, issurface=None):
+        """Parses a string to find out which brain structure is being described
+
+        :param value: string to be parsed
+        :param issurface: defines whether the object describes the volume or surface of the brain structure (default: surface if the brain structure is the cortex volume otherwise)
+        """
+        if '_' in value:
+            items = [val.lower() for val in value.split('_')]
+            if items[-1] in ['left', 'right', 'both']:
+                hemisphere = items[-1]
+                others = items[:-1]
+            elif items[0] in ['left', 'right', 'both']:
+                hemisphere = items[0]
+                others = items[1:]
+            else:
+                hemisphere = 'both'
+                others = items
+            if others[0] in ['nifti', 'cifti', 'gifti']:
+                others = others[2:]
+            primary = '_'.join(others)
+        else:
+            low = value.lower()
+            if 'left' == low[-4:]:
+                hemisphere = 'left'
+                primary = low[:-4]
+            elif 'right' == low[-5:]:
+                hemisphere = 'right'
+                primary = low[:-5]
+            elif 'both' == low[-4:]:
+                hemisphere = 'both'
+                primary = low[:-4]
+            else:
+                hemisphere = 'both'
+                primary = low
+        if issurface is None:
+            issurface = primary == 'cortex'
+        if primary == '':
+            primary = 'all'
+        return cls(primary, None, hemisphere, 'surface' if issurface else 'volume')
+
+    @classmethod
+    def from_gifti(cls, gifti_obj):
+        """
+        Extracts the brain structure from a GIFTI object
+        """
+        primary_str = 'AnatomicalStructurePrimary'
+        secondary_str = 'AnatomicalStructureSecondary'
+        primary = "unknown"
+        secondary = None
+        for meta in [gifti_obj] + gifti_obj.darrays:
+            if primary_str in meta.meta.metadata:
+                primary = meta.meta.metadata[primary_str]
+            if secondary_str in meta.meta.metadata:
+                secondary = meta.meta.metadata[secondary_str]
+        anatomy = cls.from_string(primary, issurface=True)
+        anatomy.secondary = None if secondary is None else secondary.lower()
+        return anatomy