The study of k-extendable graphs arises naturally, not only as a generalization of Sumner's randomly matchable idea, but also in the decomposition theory of graphs with respect to their maximum matchings. Central to this theory is the idea of a bicritical graph. A graph G is said to be bicritical if G-u-v has a perfect matching for all choices of vertices u,v in V(G), u != v. Bicritical graphs play a central role in the decomposition theory mentioned above and their structure is far from completely understood. It was shown in 1980 that every k-extendable graph is also (k-1)-extendable and that any 2-extendable graph is either bipartite or bicritical. Thus for non-bipartite graphs, if G_k denotes the class of those which are k-extendable and G_b those which are bicritical, we have a nested sequence of graph classes:
G_1 contains G_b contains G_2 contains G_3 contains ....
Moreover, it is known that each of the containments is proper.
We will attempt to survey the considerable amount of recent work that has been done in the area of matching extension.