„Which topologies can be shaped such that they get positively curved“ is a question which has been much investigated since the beginnings of global differential geometry. More precisely, we want to know which manifolds can be given a Riemannian metric with positive curvature. The answer depends very much on what exactly we mean be curvature. In this talk we will study the question for the weakest of these conditions, namely for scalar curvature. Starting from classical results we will see that the answer changes completely if we allow the manifold to have a nonempty boundary. There are quite a few natural boundary conditions which complement the positivity of curvature in the interior. We will see that many of them, but not all, are equivalent in a sense to be explained. The talk is based on joint work with Bernhard Hanke.