Turning a cube into a rotating multi-coloured puzzle isn't terribly difficult. Ernő Rubik did it back in 1974 without the need for a computer. For other shapes, though, like a complicated 3D bunny, you need to figure out the perfect way to slice it up so that every sub-section can rotate freely. But thankfully there's now software that can automatically do that for you.
Developed by Timothy Sun and Changxi Zheng at Columbia University, the software first requires you to have a 3D CG model of whatever object or shape you want to convert to a Rubik's-like puzzle. The software allows the user to specify where and how the 3D model will be sliced up, which will end up defining how the parts in the final puzzle rotate against each other. The software will then automatically adjust the position of those suggested slices to ensure that every section can rotate with minimal collisions. It will even go so far as to adjust the 3D model itself if a certain part poses problems, like shrinking a bunny's ears, or changing the angle of a monster's tail.
Once an object has been sliced up into a potential puzzle, the software then converts all of the pieces into actual parts, complete with hinges and grooves, so that they can actually be manufactured and assembled. To start churning out your own custom puzzles and give Rubiks Brand some competition, all you really need is a copy of the software, and access to a 3D printer.