  We consider the problem of studying the set of curvature functions which a
given compact and non-compact manifold with nonempty boundary can possess.
First we prove that the sign demanded by the Gauss-Bonnet Theorem is a
necessary and sufficient condition for a given function to be the geodesic
curvature or the Gaussian curvature of some conformally equivalent metric. Our
proof conceptually differs from Cruz-Vit\'orio's proof since our approach allow
us to solve problems where the conformal method cannot solve. Also, we prove
new existence and nonexistence of metrics with prescribed curvature in the
conformal setting which depends on the Euler characteristic. After this, we
present a higher order analogue concerning scalar and mean curvatures on
compact manifolds with boundary. We also give conditions for Riemannian
manifolds not necessarily complete or compact to admit positive scalar
curvature and minimal boundary, without any auxiliary assumptions about its
infinity, which is an extension of those proved by Carlotto-Li
