Fuchsian groups in SO(2,1) can be deformed in two basic ways: either in SO(3,1) (quasifuchsian), or in SO(2,2) (anti-de Sitter). I will give an overview of the striking parallels between the two deformation theories and their open problems. Then, I will explain a unified treatment of the deformations of punctured-torus groups: in this case, the convex core has an (infinite) triangulation that turns out to be canonical in several independent ways. This is joint work with Sara Maloni.