The Johnson homomorphism is a tool developed by Dennis Johnson to help study the mapping class group. I will define the mapping class group and give something of a survey of the major results. In particular I will define the Torelli group, a subgroup both of great importance and great intrigue in this area, and I will go over a theorem of Dehn and Nielson that embeds the mapping class group in the automorphism group of a free group. This relationship between mapping class groups and automorphisms groups has proved extremely fruitful, with results and techniques from one area frequently motivating research in the other. The Johnson homomorphism is an example of this bridge, and I will actually define it first for $Aut(F_n)$. Time permitting I will discus the recent papers Jim Conant, Martin Kassabov and Karen Vogtmann in which their hairy graph homology theory is applied to the study of the Johnson homomorphism of the mapping class group.