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Constant mean curvature (CMC) surfaces are critical points for the area functional, subject to an enclosed volume constraint. Classical examples include spheres and cylinders. Until the late 1980's the only other known examples were the Wente torus and the rotationally symmetric surfaces of Delaunay. In this talk, I will introduce CMC surfaces from a variational point of view and describe how the variational definition coupled with isometries of the ambient manifold produce conserved quantities. These conservation laws play a crucial role in the gluing techniques that I will discuss in the seminar that follows.