We review the de Branges theory of Hilbert spaces of entire functions. This theory gives a canonical form for a class of operators as multiplication operator together with a generalized Fourier transform taking such an operator to a generalized differential operator. We discuss its relation to other theories of canonical forms for certain non-self adjoint operators, including "model spaces" and Lax-Phillips scattering theory. We present examples, including de Branges spaces associated to automorphic L-functions, and discuss how the Riemann hypothesis may be encoded in this framework.