  The theory of complete surfaces of (nonzero) constant mean curvature in
$\RR^3$ has progressed markedly in the last decade. This paper surveys a number
of these developments in the setting of Alexandrov embedded surfaces; the focus
is on gluing constructions and moduli space theory, and the analytic techniques
on which these results depend. The last section contains some new results about
smoothing the moduli space and about CMC surfaces in asymptotically Euclidean
manifolds.
