If S is a compact surface, the classical Gauss-Bonnet theorem asserts that a certain topological invariant (the Euler characteristic) can be computed by integrating the Gauss curvature of any metric g over S. This integral has a natural generalization to higher dimensions called the "total scalar curvature". The total scalar curvature, however, depends very strongly on the choice of metric in all dimensions > 2. Nonetheless, the associated variational problem does allow one to attach a certain real number Y(M), called the "Yamabe invariant" or "sigma constant", to each smooth, compact manifold M. Until recently, however, there were no non-trivial examples where Y(M) could actually be computed. Fortunately, this situation has now changed decisively. In particular, we will see how Y(M) can typically distinguish between different smooth structures on a fixed topological 4-manifold. Indeed, some of the most dramatic consequences of Donaldson theory and Seiberg-Witten theory can now be sharpened into concrete assertions concerning the precise values of Yamabe invariants.