Constant mean curvature (CMC) surfaces are critical points to the area functional with an enclosed volume constraint. Classic examples include the round sphere and a one parameter family of rotationally invariant surfaces discovered by Delaunay. In this talk I outline a generalized gluing method we develop that produces infinitely many new examples of embedded CMC surfaces of finite topology. In particular, I explain how we solve the global linearized problem in the presence of possible obstructions and how we handle the remaining higher order terms. Finally, I will mention aspects of the proof we must alter to adapt the method to higher dimensions. This work is joint with Nicos Kapouleas.