  We review the paradigm of quintom cosmology. This scenario is motivated by
the observational indications that the equation of state of dark energy across
the cosmological constant boundary is mildly favored, although the data are
still far from being conclusive. As a theoretical setup we introduce a no-go
theorem existing in quintom cosmology, and based on it we discuss the
conditions for the equation of state of dark energy realizing the quintom
scenario. The simplest quintom model can be achieved by introducing two scalar
fields with one being quintessence and the other phantom. Based on the
double-field quintom model we perform a detailed analysis of dark energy
perturbations and we discuss their effects on current observations. This type
of scenarios usually suffer from a manifest problem due to the existence of a
ghost degree of freedom, and thus we review various alternative realizations of
the quintom paradigm. The developments in particle physics and string theory
provide potential clues indicating that a quintom scenario may be obtained from
scalar systems with higher derivative terms, as well as from non-scalar
systems. Additionally, we construct a quintom realization in the framework of
braneworld cosmology, where the cosmic acceleration and the phantom divide
crossing result from the combined effects of the field evolution on the brane
and the competition between four and five dimensional gravity. Finally, we
study the outsets and fates of a universe in quintom cosmology. In a scenario
with null energy condition violation one may obtain a bouncing solution at
early times and therefore avoid the Big Bang singularity. Furthermore, if this
occurs periodically, we obtain a realization of an oscillating universe.
Lastly, we comment on several open issues in quintom cosmology and their
connection to future investigations.
