  We prove compactness of solutions to some fourth order equations with
exponential nonlinearities on four manifolds. The proof is based on a refined
bubbling analysis, for which the main estimates are given in integral form. Our
result is used in a subsequent paper to find critical points (via minimax
arguments) of some geometric functional, which give rise to conformal metrics
of constant $Q$-curvature. As a byproduct of our method, we also obtain
compactness of such metrics.
