  We prove the existence of metrics with prescribed $Q$-curvature under natural
assumptions on the sign of the prescribing function and the background metric.
In the dimension four case, we also obtain existence results for curvature
forms requiring only restrictions on the Euler characteristic. Moreover, we
derive a prescription result for open submanifolds which allow us to conclude
that any smooth function on $\mathbb{R}^n$ can be realized as the $Q$-curvature
of a Riemannian metric.
