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

Existence and obstructions for the curvature on manifolds with boundary