After a brief historical introduction we will review the more recent developments in the subject. These include the construction of constant mean curvature tori by Wente, the understanding of the embedded case achieved by the work of Meeks, Korevaar, Kusner, and Solomon, and classification results for tori by Pinkall, Sterling and others.
The main part of the lecture will concentrate on the methods developed to construct nontoroidal constant mean curvature surfaces. The interplay between Geometry and Analysis will be emphasized and the fundamental ideas of the approach developed will be described. Subsequently we will discuss the technical realization of these ideas and the idiosyncracies of applying then in the cases at hand.
In the final part of the lecture we will outline the open problems in the field. We will also discuss the analogies and the applicability of the methods previously described to other geometric problems.