In this talk I'll describe recent joint work with Xin Zhou, where we make progress on the question of finding closed constant mean curvature surfaces with controlled topology in 3-manifolds. We show that in a 3-sphere equipped with an arbitrary Riemannian metric, there exists a branched immersed 2-sphere with constant mean curvature H for almost every H. Moreover, the existence extends to all H when the target metric is positively curved. This latter result confirms, for the branched immersed case, a conjecture of Harold Rosenberg and Graham Smith.