A graph is a "minor" of another if it can be obtained from a subgraph of the second by contracting edges. Perhaps the most well-known open question about minors is Hadwiger's conjecture, extending the four-color theorem, that for all t, every graph with no K_{t+1} minor can be t-colored. After an introduction to minors of graphs, the course focuses on aspects of Hadwiger's conjecture, particularly on recent results giving upper bounds on the chromatic number of graphs with no K_{t+1} minor.