The moduli space (stack) $\cM_{g,n}$ of complex structures on an $n$-pointed Riemann surface of genus $g,$ for $2g+n\ge3,$  carries natural K\"ahler metrics with the Weil-Petersson metric of particular significance. I will describe what is known about the asymptotic behaviour of the metric and how this can be used to show that the Hodge theory, in the sense of the square-integrable harmonic forms, can be identified with the cohomology of the Knudsen-Deligne-Mumford compactification.