  The Gromov-Lawson-Rosenberg-conjecture for a group G states that a closed
spin manifold M^n (n>4) with fundamental group G admits a metric with positive
scalar curvature if and only if its C^*-index A(M) in KO_n(C^*_r(G)) vanishes.
We prove this for groups G with low-dimensional classifying space, provided the
assembly map for G is injective.
  On the other hand, we construct a spin manifold with no metric with scal>0
but so that already its KO-orientation in KO_*(B pi_1(M)) vanishes. Therefore a
corresponding weakened version or the GLR-conjecture is wrong.
  Last we address non-orientable manifolds. We give a reformulation of the
minimal surface method of Schoen and Yau (extended to dimension 8) and
introduce a non-orientable version of it. We then construct a manifold M whose
orientation cover admits a metric of positive scalar curvature but where M
itself does not. M is also a counterexample to a twisted analog of the
GLR-conjecture because its twisted index vanishes.
