The Helicopter Cube is a puzzle designed by Adam Cowan, a designer of a number of interesting puzzle including the Axis cube and Jing's Pyraminx. The puzzle is cubic with 8 corners. Each face is divided into 4 centre-pieces. There are 3 centre-pieces around each corner.
This is one of the most unusual puzzles I have seen. When you first have a go with one, you turn 2 adjacent corners along with 4 of the centre pieces. This allows corners to move and rotate and for the edges to be jumbled. This makes 12 different turns and an interesting enough challenge. It's relatively straight forward to solve half of the puzzle with intuition alone. For the last layer, you'll probablly get there on your own with enough patience, fairly quickly if you look for one or two algorithms to help you with the last few pieces.
If you turn one of the corner-pairs about 70° or so, you can make a turn with another corner-pair. Now you have a shape-changing puzzle. Worse still, if you restore the shape, there's a pretty good chance the pieces have been placed in a position that means you can't solve it without changing the shape.