  We survey known (and unknown) results about the behavior of Heegaard genus of
3-manifolds constructed via various gluings. The constructions we consider are
(1) gluing together two 3-manifolds with incompressible boundary, (2) gluing
together the boundary components of surface times I, and (3) gluing a
handlebody to the boundary of a 3-manifold. We detail those cases in which it
is known when the Heegaard genus is less than what is expected after gluing.
