Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of . 1. Introduction In recent years, much attention has been devoted to the study of partial diﬀerential equations on manifolds, in order to understand some connections between analytic and geometric properties of these objects. .