The ordinary game of Nim has a long history and is well-known in the area of combinatorial game theory. The solution to the ordinary game of Nim has been known for many years and lends itself to numerous other solutions to combinatorial games. Nim was extended to graphs by taking a fixed graph with a playing piece on a given vertex and assigning positive integer weight to the edges that correspond to a pile of stones in the ordinary game of Nim. Players move alternately from the playing piece across incident edges, removing weight from edges as they move. Few results in this area have been found, leading to its appeal. This dissertation examines broad classes of graphs in relation to the game of Nim to find winning strategies and to solve the problem of finding the winner of a game with both unit weighting assignments and with arbitrary weighting assignments. Such classes of graphs include the complete graph, the Petersen graph, hypercubes, and bipartite graphs. We also include the winning strategy for even cycles.