The mapping class group of a Heegaard splitting for a 3-manifold is the group of isotopy classes of self-homeomorphisms of the manifold preserving the splitting. It is natural to try to understand the structure of each of those groups and to find a reasonable generating set or a presentation of it. In this talk, we review the history of this problem and introduce some recent progress on it. In particular, we will look at finite presentations of the groups of reducible, genus two Heegaard splittings, and the key ideas to obtain them. This is a joint work with Yuya Koda.