This talk presents a survey about what is known concerning the question which compact manifolds admit metrics of positive scalar curvature. The central conjecture in the subject is the Gromov-Lawson-Rosenberg conjecture which claims that a spin manifold $M$ of dimension $n\ge 5$ admits a positive scalar curvature metric if and only if an ``index obstruction" $\alpha(M)\in KO_n(C^*\pi)$ vanishes. Here $\pi$ is the fundamental group of $M$, and $KO_n(C^*\pi)$ is the $K$-theory of the $C^*$-algebra of $\pi$ (a norm completion of the real group ring of $\pi$). This conjecture has been proved for manifolds whose fundamental group has periodic cohomology, but recently a counterexample was found for $\pi_1(M)=\Bbb Z^4\times \Bbb Z/3$. However, a weaker version of the Conjecture has been proved for all manifolds with finite fundamental groups, and more recently, for all manifolds for whose fundamental group a Conjecture a la Novikov holds (this form of the ``Novikov-Conjecture" has e.g. been verified for discrete subgroups of Lie groups).