I'm sure this puzzle has a name that it's called for commercial purposes, but I can't remember what it is, nor is it easy to figure out if you have no clue what search terms to use. I'm just going to call it the round-ish cube.
At first glance, this puzzle doesn't look terribly interesting, but I think it is. You clearly solve this with any sequence you already know for the classic 3x3x3 Rubik's cube. You quickly find, however, that you often run into a dead-end where the solve just doesn't go through. Furthermore, there appears to actually be more than one solved state of the puzzle, if by solved we mean that each face is a solid color. There is, perhaps, though, just one solved state where each color involved appears in pairs where each pair is on opposite sides of the cube. In other cases, just a few such pairs are opposite one another while the rest are adjacent.
Perhaps the only useful thing I can say about this puzzle is that if you have it all solved except for two face pieces swapped, then to resolve this, you just need to swap two face pieces of the same color. This is a bit of a back-track, but once done, the solve should go through.