Many partial differential equations (PDEs) that arise in physics can be viewed as infinite dimensional Hamiltonian systems. This monograph presents recent existence rsults of nonlinear oscillarions of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations. After introducing the reader to classical finite-dimensional dynamical system theory, including the Weinstein-Moser and Fadell-Rabinowitz center theorems, an analogous theory for completely resonant nonlinear wave equations is developed. Within this theory both problems of small divisors and infinite bifurcation phenomena occur, requiring the use of Nash-moser theory as well as minimax variational methods. These techinques are presented in a self contained manner toghether with other basic notions of Hamiltonian PDEs and number theory.
