  In this review we summarize the effect of relative supersonic velocities
between gas and dark matter created right after recombination on astrophysics
and cosmology. The relative velocities formally introduce a second order effect
on the standard results and thus have been neglected in the framework of linear
theory. However, when properly considered, the velocities yield a
non-perturbative contribution to the growth of structures which is then
inherited by the majority of cosmic signals coming from redshifts above z ~ 10,
and in certain cases may even propagate to various low-redshift observables
such as the Baryon Acoustic Oscillations measured from the distribution of
galaxies. At higher redshifts, the supersonic velocities have thus strong
impact affecting the abundance of first star forming halos in an inhomogeneous
way, hindering creation of the first stars, leaving traces in the redshifted
21-cm signal of neutral hydrogen, as well as having other important
contributions at high redshifts all of which we review in this manuscript.
