  We show that in the semiclassical limit, classically chaotic systems have
universal spectral statistics. Concentrating on short-time statistics, we
identify the pairs of classical periodic orbits determining the small-$\tau$
behavior of the spectral form factor $K(\tau)$ of fully chaotic systems. The
two orbits within each pair differ only by their connections inside close
self-encounters in phase space. The frequency of occurrence of these
self-encounters is determined by ergodicity. Permutation theory is used to
systematically sum over all topologically different families of such orbit
pairs. The resulting expansions of the form factor in powers of $\tau$ coincide
with the predictions of random-matrix theory, both for systems with and without
time-reversal invariance, and to all orders in $\tau$. Our results are closely
related to the zero-dimensional nonlinear $\sigma$ model of quantum field
theory. The relevant families of orbit pairs are in one-to-one correspondence
to Feynman diagrams appearing in the perturbative treatment of the $\sigma$
model.
