  Bouncing cosmologies, suggested by String/M-theory, may provide an
alternative to standard inflation to account for the origin of inhomogeneities
in our universe. The fundamental question regards the correct way to evolve the
scalar perturbations through the bounce. In this work, we determine the
evolution of perturbations and the final spectrum for an arbitrary (spatially
flat) bouncing cosmology, with the only assumption that the bounce is governed
by a single physical scale. In particular, we find that the spectrum of the
pre-bounce growing mode of the Bardeen potential (which is scale-invariant in
some limit, and thus compatible with observations) survives unaltered in the
post-bounce only if the comoving pressure perturbation is directly proportional
to the Bardeen potential rather than its Laplacian, as for any known form of
ordinary matter. If some new physics acting at the bounce justifies such
relation, then bouncing cosmologies are entitled to become a real viable
alternative for the generation of the observed inhomogeneities. Our treatment
also includes some class of models with extra-dimensions, whereas we show that
bounces induced by positive spatial curvature are structurally different from
all bounces in spatially flat universes, requiring a distinct analysis.
