  The Bohigas--Giannoni--Schmit conjecture stating that the statistical
spectral properties of systems which are chaotic in their classical limit
coincide with random matrix theory is proved. For this purpose a new
semiclassical field theory for individual chaotic systems is constructed in the
framework of the non--linear $\sigma$-model. The low lying modes are shown to
be associated with the Perron--Frobenius spectrum of the underlying
irreversible classical dynamics. It is shown that the existence of a gap in the
Perron-Frobenius spectrum results in a RMT behavior. Moreover, our formalism
offers a way of calculating system specific corrections beyond RMT.
