Abstract: This talk, based on work at the summer 2006 REU at Canisius, will explain the idea of groups from Abstract Algebra and directed graphs from Discrete Math, concentrating on some interesting examples. A group G acting on a set is called a G-set. A G-set together with a list of elements in G gives a directed graph, which I call an action graph; Cayley graphs are a special case. This talk will explore all action graphs for a fixed group, the symmetric group S4. Applications include the symmetries of the cube, characteristic polynomials, and Burnside-Polya counting.