  We study random-matrix ensembles with a non-Gaussian probability distribution
$P(H) \sim \exp (-N {\rm tr }\, V(H))$ where $N$ is the dimension of the matrix
$H$ and $V(H)$ is independent of $N$. Using Efetov's supersymmetry formalism,
we show that in the limit $N \rightarrow \infty$ both energy level correlation
functions and correlation functions of $S$-matrix elements are independent of
$P(H)$ and hence universal on the scale of the local mean level spacing. This
statement applies to each of the three generic ensembles (unitary, orthogonal,
and symplectic). Universality is also found for correlation functions depending
on some external parameter. Our results generalize previous work by Brezin and
Zee [Nucl.\ Phys.\ B {\bf 402}, 613 (1993)].
