  Let S be a compact oriented surface. A homology cobordism of S is a cobordism
C between two copies of S, such that both the "top" inclusion and the "bottom"
inclusion of S in C induce isomorphisms in homology. Homology cobordisms of S
form a monoid, into which the mapping class group of S embeds by the mapping
cylinder construction. In this paper, we survey recent works on the structure
of the monoid of homology cobordisms, and we outline their relations with the
study of the mapping class group. We are mainly interested in the cases where
the boundary of S is empty or connected.
