  This is a survey on the global theory of constant mean curvature surfaces in
Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight
canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2
\times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie
group Sol3. We will focus on the problems of classifying compact CMC surfaces
and entire CMC graphs in these spaces. A collection of important open problems
of the theory is also presented.

Differential Geometry