  We explore simple but novel bouncing solutions of general relativity that
avoid singularities. These solutions require curvature k=+1, and are supported
by a negative cosmological term and matter with -1 < w < -1/3. In the case of
moderate bounces (where the ratio of the maximal scale factor $a_+$ to the
minimal scale factor $a_-$ is ${\cal O}(1)$), the solutions are shown to be
classically stable and cycle through an infinite set of bounces. For more
extreme cases with large $a_+/a_-$, the solutions can still oscillate many
times before classical instabilities take them out of the regime of validity of
our approximations. In this regime, quantum particle production also leads
eventually to a departure from the realm of validity of semiclassical general
relativity, likely yielding a singular crunch. We briefly discuss possible
applications of these models to realistic cosmology.
