  We explicitly compute the diverging factor in the large genus asymptotics of
the Weil-Petersson volumes of the moduli spaces of $n$-pointed complex
algebraic curves. Modulo a universal multiplicative constant we prove the
existence of a complete asymptotic expansion of the Weil-Petersson volumes in
the inverse powers of the genus with coefficients that are polynomials in $n$.
This is done by analyzing various recursions for the more general intersection
numbers of tautological classes, whose large genus asymptotic behavior is also
extensively studied.
