We will consider the classical problem, proposed by Kazdan and Warner in the 70’s, of prescribing the scalar curvature of a Riemannian manifold via conformal deformations of the metric. This amounts to solving an elliptic nonlinear PDE with critical exponent, presenting difficulties due to lack of compactness. There are in general obstructions, but still several contributions to the existence theory have been given using different techniques. These include direct methods of the calculus of variations, blow-up analysis, Liouville theorems, gluing constructions and topological or Morse-theoretical tools. We will give a general presentation of the subject, describing the principal contributions in the literature and arriving to more recent developments.