This work is devoted to the case of constant mean curvature surfaces immersed in R (or, more generally, in spaces of constant curvature). Wente reduces this geometrical problem to finding certain integrable solutions to the Gauss equation. Many new and interesting examples are presented, including immersed cylinders in R with embedded Delaunay ends and n-lobes in the middle, and one-parameter families of immersed cmc tori in R . Finally, Wente examines minimal surfaces in hyperbolic three-space, which is in some ways the most complicated case.