  We present accurate Monte-Carlo calculations of Lyman Alpha (hereafter Lya)
radiation pressure in a range of models which represent galaxies during various
epochs of our Universe. We show that the radiation force that Lya photons exert
on hydrogen gas in the neutral intergalactic medium (IGM), that surrounds
minihalos that host the first stars, may exceed gravity by orders of magnitude
and drive supersonic winds. Lya radiation pressure may also dominate over
gravity in the neutral IGM that surrounds the HII regions produced by the first
galaxies. However, the radiation force is likely too weak to result in
supersonic outflows in this case. Furthermore, we show that Lya radiation
pressure may drive outflows in the interstellar medium of star forming galaxies
that reach hundreds of km/s. This mechanism could also operate at lower
redshifts z<6, and may have already been indirectly detected in the spectral
line shape of observed Lya emission lines.
