Assuming your interest is in constant mean curvature surfaces with circular boundary, I found the survey, <a href="http://www.ugr.es/~rcamino/publications/pdf/art57.pdf">"Surfaces with constant mean curvature in Euclidean space"</a> by R. Lopez to be a great introduction, and it contains the state of the art, and several references. 

Hopf proved that the only constant mean curvature closed surfaces of genus 0 are spheres. Alexandrov's theorem says that constant mean curvature closed surfaces that are embedded are spheres. Unfortunately when we allow boundaries both analogs are conjectural:

>__Conjecture 1:__ The only constant mean curvature compact surfaces with circular boundary that are topological disks are spherical caps.
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>__Conjecture 2:__ The only constant mean curvature compact surfaces with circular boundary that are embedded are spherical caps.