  Several decades of observations and discoveries have shown that high-redshift
AGN and massive galaxies are often surrounded by giant Lyman-alpha nebulae
extending in some cases up to 500 kpc in size. In this review, I discuss the
properties of the such nebulae discovered at z>2 and their connection with gas
flows in and around the galaxies and their halos. In particular, I show how
current observations are used to constrain the physical properties and origin
of the emitting gas in terms of the Lyman-alpha photon production processes and
kinematical signatures. These studies suggest that recombination radiation is
the most viable scenario to explain the observed Lyman-alpha luminosities and
Surface Brightness for the large majority of the nebulae and imply that a
significant amount of dense, ionized and cold clumps should be present within
and around the halos of massive galaxies. Spectroscopic studies suggest that,
among the giant Lyman-alpha nebulae, the one associated with radio-loud AGN
should have kinematics dominated by strong, ionized outflows within at least
the inner 30-50 kpc. Radio-quiet nebulae instead present more quiescent
kinematics compatible with stationary situation and, in some cases, suggestive
of rotating structures. However, definitive evidences for accretion onto
galaxies of the gas associated with the giant Lyman-alpha emission are not
unambiguously detected yet. Deep surveys currently ongoing using other bright,
non-resonant lines such as Hydrogen H-alpha and HeII1640 will be crucial to
search for clearer signatures of cosmological gas accretion onto galaxies and
AGN.
