  In this paper we study the problem of finding a conformal metric with the
property that the k-th elementary symmetric polynomial of the eigenvalues of
its Weyl-Schouten tensor is constant. A new conformal invariant involving
maximal volumes is defined, and this invariant is then used in several cases to
prove existence of a solution, and compactness of the space of solutions
(provided the conformal class admits an admissible metric). In particular, the
problem is completely solved in dimension four, and in dimension three if the
manifold is not simply connected.
