In this chapter we begin our study of the mapping class group of a surface. After giving the definition, we compute the mapping class group in essentially all of the cases where it can be computed directly. This includes the case of the disk, the annulus, the torus, and the pair of pants. An important method, which we call the Alexander method, emerges as a tool for such computations. It answers the fundamental question: how can one prove that a homeomorphism is or is not homotopically trivial? Equivalently, how can one decide when two homeomorphisms are homotopic or not?