We will discuss the following question: is it possible to find a
Riemannian metric with given Ricci curvature on a manifold $M$? To
answer this question, one must analyze a weakly elliptic
second-order partial differential equation for tensors. In the first
part of the talk, we will review the relevant background and the
history of the subject. After that, our focus will be on new results
concerning the case where $M$ is a bounded domain in a cohomogeneity
one manifold.