  Gravitational waves (GWs) have a great potential to probe cosmology. We
review early universe sources that can lead to cosmological backgrounds of GWs.
We begin by presenting definitions of GWs in flat space-time and in a
cosmological setting, and discussing the reasons why GW backgrounds from the
early universe are of a stochastic nature. We recap current observational
constraints on stochastic backgrounds, and discuss some of the characteristics
of present and future GW detectors including advanced LIGO, advanced Virgo, the
Einstein Telescope, KAGRA, LISA. We then review in detail early universe GW
generation mechanisms proposed in the literature, as well as the properties of
the GW backgrounds they give rise to. We classify the backgrounds in five
categories: GWs from quantum vacuum fluctuations during standard slow-roll
inflation, GWs from processes that operate within extensions of the standard
inflationary paradigm, GWs from post-inflationary preheating and related
non-perturbative phenomena, GWs from first order phase transitions (related or
not to the electroweak symmetry breaking), and GWs from topological defects, in
particular from cosmic strings. The phenomenology of early universe processes
that can generate a stochastic background of GWs is extremely rich, and some
backgrounds are within the reach of near-future GW detectors. A future
detection of any of these backgrounds will provide crucial information on the
underlying high energy theory describing the early universe, probing energy
scales well beyond the reach of particle accelerators.
