Since "cubes" with higher dimension than three exist I think it's natural to ask for higher dimensional Rubik's cubes. These so called hypercubes don't seem to have been described from a group theoretic point of view.

Are there any papers on this? Is the group of the $3\times 3\times 3 \times 3$ cube a subgroup of a wreath product of another wreath product?

In case you don't know about the $3\times 3\times 3$ cube. Its group is a subgroup of a direct group of wreath products. The wreath products describing the corner pieces and the edge pieces and representing a permutation of them with its action in the respective orientation. That is why I conjecture that in the $3\times 3\times 3 \times 3$ case we might get a wreath product of the permutation of the faces, which are now 3d cubes itself, by the the wreath product of the permutation of the 2d faces of these by the groups of their orientations.