Conformal metrics and prescription of scalar curvature

We consider the classical problem of prescribing the scalar curvature of a manifold, dating back to works by Kazdan and Warner. On the sphere, when one uses conformal metrics, this is also known as Nirenberg's problem. This problem is not always solvable: one of the difficulties arising when studying it is the loss of compactness, mainly caused by the action of the Möbius group and by the criticality of Sobolev's embeddings. This phenomenon, referred to as "bubbling" is mainly understood in low dimensions, where bubbles must be either single or at finite distance from each other. We will review some history on the problem, discussing the main ideas involved to tackle the above issues. We will also present some recent work with M.Mayer concerning existence and non-existence results of new type for the higher-dimensional case.