The mapping class group of a Heegaard splitting for a given 3-manifold is the group of automorphisms of the 3-manifold that take the Heegaard surface onto itself, modulo isotopies that preserve the surface setwise. This can be viewed as a subgroup of the mapping class group of the surface. I will discuss a number of equivalent definitions of this group and describe some recent examples that demonstrate an interesting relationship between the structure of the mapping class group and the topology of the ambient 3-manifold.