We consider a class of Liouville equations that arise in differential geometry when prescribing the Gaussian curvature of a surface and in models of mathematical physics describing stationary Euler flows and self-dual Chern-Simons equations. We discuss methods, variational in nature, to derive general existence results from suitable improvements of the Moser-Trudinger inequality combined with Morse-theoretical methods. We will treat in particular the case with Dirac masses representing, in the above motivations, conical singularities or vortex points.