I'm no expert in this field, but I am familiar with the classification of rotationally symmetric surfaces with constant mean curvature by Delaunay. I am aware that once we drop embeddedness and smoothness, the situation becomes very complicated.
My question is, other than the family of surfaces from Delaunay, is there a classification of (embedded in a Riemannain manifold $M^n$, smooth) surfaces with constant mean curvature? If not, is there a classification known under additional conditions? $M^n = R^n$, closed surfaces, surfaces with boundary, finite total curvature, etc?
I write such a broad question as I find this topic quite interesting and beautiful, and hope that others also share this feeling.