  Let $M$ be a simply-connected closed manifold of dimension $\geq 5$ which
does not admit a metric with positive scalar curvature. We give necessary
conditions for $M$ to admit a scalar-flat metric. These conditions involve the
first Pontrjagin class and the cohomology ring of $M$. As a consequence any
simply-connected scalar-flat manifold of dimension $\geq 5$ with vanishing
first Pontrjagin class admits a metric with positive scalar curvature. We also
describe some relations between scalar-flat metrics, almost complex structures
and the free loop space.
