The setting of this paper is four dimensional, emphasizing certain parallels between conformal geometry in two and four dimensions. The analogy is illustrated by discussing the Gauss-Bonnet formula for compact surfaces and the Chern-Gauss-Bonnet integrand in four dimensions. This leads to the conjecture that, on a compact four-manifold, the positivity of conformal invariants and the Yamabe invariant implies the existence of a conformal metric with strictly positive Ricci curvature. The conjecture is resolved in the affirmative way. The proof involves the study of a fully nonlinear equation of Monge-Ampère type and the application of heat equation techniques.