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Although the inflationary paradigm is the most widely accepted explanation for the current cosmological observations, it does not necessarily correspond to what actually happened in the early stages of our Universe. To decide on this issue, two paths can be followed: first, all the possible predictions it makes must be derived thoroughly and compared with available data, and second, all the imaginable alternatives must be ruled out. Leaving the first task to all other contributors of this volume, we concentrate here on the second option, focusing on the bouncing alternatives and their consequences. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]