  This is a survey of the current state of the question "Which closed connected
manifolds of dimension $n\ge 5$ admit Riemannian metrics whose scalar curvature
function is everywhere positive?" The introduction gives a brief overview of
these results, while the body of the paper discusses the methods used in the
proofs of these results. We mention the two flavors of topological obstructions
to the existence of \pscm s: one is a consequence of the Weizenb\"ock formula
for the Dirac operator, the other is obtained by considering stable minimal
hypersurfaces. We talk about geometric constructions of \pscm s (the
surgery/bordism theorem), which shows that the answer to the question above
depends only the bordism class of the manifold in a suitable bordism group. Via
the Pontryagin-Thom construction this can be translated into stable homotopy
theory, and solved completely in some cases, in particular for simply connected
manifolds, or manifolds with very special fundamental groups. The last section
discusses some open questions.
