Master Pyramorphix
This beautiful pillowed tetrahedron is not as it first appears: one might expect the axes of rotation to be similar to the Pyraminx but a quick play with the puzzle has most people exclaiming "Wow! This is wierd!" as it's a shape-shifter.
The puzzle comes in a single colour version and a four-colour version. I have a Meffert's 4-colour model with a white body and flourescent stickers.
See http://en.wikipedia.org/wiki/Pyramorphix#Master_Pyramorphix for some technical information.
Regarding the name: Meffert's refers to this puzzle as a "Master Pyramorphinx" with an "N" thrown in there - I'm not sure why but I'll find out soon probably!
The Master Pyramorphix is based on the 3x3x3 mechanism which defies logic for a while as one gets used to the way the pieces move. It can be solved as a 3x3x3 for the most part but some consideration needs to be given to the rotation of the "centres", which lie on the midpoint of each edge of the tetrahedron, as each one has two stickers. Also only half of the eight "corners" have a correct orientation when solved: they have two coloured stickers whereas the remaining four, the little triangular pieces in the centre of each face, have only one sticker and one orientation. We also see some "parity" cases with opposite corners requiring a swap and this arrises from the lack of orentation of half of the "corner" pieces.
At the moment (and I've only had the puzzle a few hours!) I'm using a layer-by-layer 3x3x3 method with a beginners' multi step last-layer. I'm happy to say that I've learned this by myself but that's no major achievement by many people's standards!