2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/224975We 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.This is the version published by Geometry & Topology Monographs on 3 December 2007Geometric Topology57M99, 57N10, 57M27text