TITLE: Is there a complete classification of constant mean curvature surfaces?
QUESTION [13 upvotes]: 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.

REPLY [8 votes]: Since my previous answer to this question major progress was made: As Robert Haslhofer already mentioned, Brendle proved the Lawson conjecture that the only minimal CMC torus in the 3-sphere is the Clifford torus. Building on this work, Andrews and Li ("Embedded constant mean curvature tori in the three-sphere." J. Differential Geom. 99 (2015)) classified all embedded CMC tori in $S^3.$ To be more concrete, they used the same two point function as in Brendles proof in order to show that every embedded CMC torus must be rotationally symmetric, and hence is classified and can be written down in terms of elliptic functions explicitly.
The case of higher genus CMC surfaces in space forms is more difficult and unsolved by now. But there has been done computer experiments which suggest that the space of embedded CMC surfaces of higher genus $g\geq2$ with certain symmetries is related to the space of embedded CMC tori, see http://arxiv.org/pdf/1503.07838.pdf and the following image: