One of the most popular puzzles ever invented, Rubik's cube, hides mathematical ideas of remarkable complexity. The 43 quintillion configurations possible on a standard 3x3x3 cube can be organized in an interesting algebraic structure, the Rubik's cube group. Using ideas and tools from abstract algebra (permutation groups, conjugations, group actions), we will describe the structure of the Rubik's cube group and outline a method for solving any configuration. We will extend these ideas to other "twisty puzzles": larger cubes, pyraminx, and megaminx.