2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/212999In this mostly expository paper we review several known results about the cohomology of moduli spaces of smooth and stable curves, focusing in particular on low degree cohomology. We also give a new proof of Harer's theorem describing the second cohomology group of the moduli space of smooth n-pointed curves of given genus18 pages, 1 figure; written for Surveys in Differential Geometry. Minor mistakes corrected, references addedAlgebraic Geometry14H10text