Constant mean curvature surfaces

In this article we survey recent developments in the theory of constant mean cur- vature surfaces in homogeneous 3-manifolds, as well as some related aspects on ex- istence and descriptive results for H -laminations and CMC foliations of Riemannian n -manifolds. Mathematics Subject Classi cation: Primary 53A10, Secondary 49Q05, 53C42. Key words and phrases: Minimal surface, constant mean curvature, minimal lamina- tion, H -lamination, CMC foliation, locally simply connected, Jacobi function, stabil- ity, index of stability, Shi man function, curvature estimates, parking garage structure, local picture on the scale of topology, Willmore energy.