  We study geometric and spectral properties of typical hyperbolic surfaces of
high genus, excluding a set of small measure for the Weil-Petersson probability
measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H
as the genus g goes to infinity. An estimate for the number of eigenvalues in
an interval [a,b] in terms of a, b and g is then proven using the Selberg trace
formula. It implies the convergence of spectral measures to the spectral
measure of H as g $\rightarrow$+$\infty$, and a uniform Weyl law as b
$\rightarrow$+$\infty$. We deduce a bound on the number of small eigenvalues,
and the multiplicity of any eigenvalue.
