Several decades of observations and discoveries have shown that highredshift AGN and massive galaxies are often surrounded by giant Lyα nebulae extending in some cases up to 500 kpc in size. In this chapter, I review the properties of the such nebulae discovered at z > 2 and their connection with gas flows in and around 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 Lyα photon production processes and kinematical signatures. The emerging picture from these studies suggest that recombination radiation is the most viable scenario to explain the observed Lyα luminosities and Surface Brightness for the large majority of the nebulae and imply that a significant amount of dense (n > 1 cm −3 ), ionized and cold (T ∼ 10 4 K) "clumps" should be present within and around the halos of massive galaxies. Spectroscopic studies suggest that, among the giant Lyα 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 Lyα emission are not unambiguously detected yet. Deep surveys currently ongoing using other bright, non-resonant lines such as Hydrogen Hα and He II 1640 will be crucial to search for clearer signatures of cosmological gas accretion onto galaxies and AGN.