Given a compact manifold M, the set S of Riemannian metrics with prescribed volume form V is totally geodesic with respect to the Ebin metric; by a classical result of Moser, S contains all possible Riemannian structures on M. In this talk, I will describe some classical results on the behaviour of the Einstein-Hilbert functional on S, and how these results have been used to construct new Einstein metrics. I will also describe a new result that applies when M is at least five-dimensional: given a Riemannian metric in S, there is an open and dense set of Ebin geodesics on S starting at g (in the smooth Whitney topology), along which scalar curvature converges to - infinity uniformly on M. This is joint work with Christoph Bohm and Brian Clark.