  Given the proliferation of bouncing models in recent years, we gather and
critically assess these proposals in a comprehensive review. The Planck data
shows an unmistakably red, quasi scale-invariant, purely adiabatic primordial
power spectrum and no primary non-Gaussianities. While these observations are
consistent with inflationary predictions, bouncing cosmologies aspire to
provide an alternative framework to explain them. Such models face many
problems, both of the purely theoretical kind, such as the necessity of
violating the NEC and instabilities, and at the cosmological application level,
as exemplified by the possible presence of shear. We provide a pedagogical
introduction to these problems and also assess the fitness of different
proposals with respect to the data. For example, many models predict a slightly
blue spectrum and must be fine-tuned to generate a red spectral index; as a
side effect, large non-Gaussianities often result.
  We highlight several promising attempts to violate the NEC without
introducing dangerous instabilities at the classical and/or quantum level. If
primordial gravitational waves are observed, certain bouncing cosmologies, such
as the cyclic scenario, are in trouble, while others remain valid. We conclude
that, while most bouncing cosmologies are far from providing an alternative to
the inflationary paradigm, a handful of interesting proposals have surfaced,
which warrant further research. The constraints and lessons learned as laid out
in this review might guide future research.
