I will give a survey of some of the exciting progress in the classical theory of surfaces M in 3-manifolds with constant mean curvature H greater than or equal to zero; we call such a surface an H-surface. The talk will cover the following topics: 1. The classification of properly embedded genus 0 minimal surfaces in R^3. (joint with Perez and Ros) 2. The theorem that for any c>0, there exists a constant K=K(c) such that for H>c, and any compact embedded H-disk D in R^3 (joint with Tinaglia): (a) the radius of D is less than K. (b) the norm of the second fundamental form of D is less than K for any points of D of intrinsic distance at least c from the the boundary of D is less than K. (c) item 2(b) works for any compact embedded H-disk (H>c) in any complete homogeneous 3-manifold with absolute sectional curvature less than 1 for the same K. 3 For c>0, there exists a constant K such that for any complete embedded H-surface M with injectivity radius greater than c>0 in a Riemannian 3-manifold with absolute sectional curvature <1 has the norm of its second fundamental form less than K. (joint with Tinaglia) (a) Complete embedded finite topology H-surfaces in R^3 have positive injectivity radius and are properly embedded with bounded curvature. (b) Complete embedded simply connected H-surfaces in R^3 are spheres, planes and helicoids; complete embedded H-annuli are catenoids and Delaunay surfaces. (c) Complete embedded simply-connected and annular H-surfaces in H^3 with H less than or equal to 1 are spheres and horospheres, catenoids and Hsiang surfaces of revolution; the key fact here is that complete + connected implies proper. 3. Classification of the conformal structure and asymptotic behavior of complete injective H-annuli f:S^1 x [0,1)--->R^3; there is a 2-parameter family of different structures for H=0. (joint with Perez when H=0) 4. Solution of the classical proper Calabi-Yau problem for arbitrary topology (even with disjoint limit sets for distinct ends!!). (joint with Ferrer and Martin)