The starting point will be Alexandrov's theorem, which states that the only (embedded) compact, constant mean curvature (CMC) hypersurfaces without boundary are spheres. Then we will discuss unduloids, which are noncompact CMC hypersurfaces closely related to both catenoids and spheres. Finally, we will outline some recent gluing constructions that patch together spheres and unduloids to produce more complex CMC hypersurfaces.