An important class of problems in geometric analysis consists in deforming the metric of a surface (or higher-dimensional manifold) in suitable ways, so that the new metric has a prescribed value of curvature. These problems, which include the uniformization, Nirenberg and Kazdan-Warner problems, are often studied via variational techniques, as I will briefly discuss. In particular I emphasize the difficulties arising from the lack of compactness and its relation to sharp inequalities.