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×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.
