Going back to algebraic reconstruction techniques (ART) in the area of computed tomography in the early 1970s, convex projection methods have been extensively used for solving complex inverse problems. We shall discuss the evolution of these methods over the last 3 decades and present their most recent extensions in the form of proximal methods. The proximity operator of a convex function is a powerful extension of the notion of a projection operator for a convex set which plays a central role in modern optimization. The proximal formalism will be shown to cover, explicitly or implicitly, a wide range of apparently unrelated approaches. Various applications will be discussed, including signal reconstruction under sparsity constraints.