A classical question in differential geometry concerns which smooth functions f can arise as Gauss curvature of a conformal metric on a 2-dim Riemannian manifold M. This amounts to solve a PDE which is the Euler-Lagrange equation of an energy functional. In this talk we will discuss about compactness issues and bubbling phenomena for this equation on surfaces of genus greater than 1 (joint work with Borer and Struwe) and on the torus.