The traditional game of Nim comprises of two players who take turns removing objects from distinct piles. The player who takes the last object is the winner. We consider the game Nim on Cayley graphs of finite groups, where the piles are located on the vertices and the number of objects in each pile is denoted as the weight of the vertex. In this version of the game, a player wins by trapping the opponent on a vertex with weight zero so he or she is unable to further reduce the weight of that vertex. We examine winning strategies for Nim on Cayley graphs of cyclic groups, dihedral groups, and the Quaternions, among others
