  We show that there exist modified theories of gravity in which the metric
satisfies second-order equations and in which the Big Bang singularity is
replaced by a cosmic bounce without violating any energy condition. In fact,
the bounce is possible even for presureless dust. We give a characterization of
such theories, which are formulated in the Palatini formalism, and discuss
their dynamics in the region near the bounce. We consider spatially flat and
non-flat homogeneous and isotropic universes.
