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Several decades of observations and discoveries have produced an extensive literature of large and giant Lyα nebulae. Depending on the technique and original target of the observation, i.e. quasars, radio-galaxies, overdense regions or "blank" fields, they have been classified in various ways and with different nomenclatures. For historical and practical reasons, I have divided them in quasar Lyα nebulae, radio-galaxy halos and Lyα Blobs (LAB) but, as discussed in section 2, their comparable volume densities, luminosities and their almost invariable association with AGN or massively star forming galaxies suggest however that these nebulae could be just apparently different manifestations of the same phenomenon.

Among the three processes associated with the production of Lyα photons (recombination radiation, "scattering" and collisional excitation; see section 3), recombination radiation currently appears as the most viable scenario to explain the observed Lyα luminosities and Surface Brightness for the large majority of the nebulae. If contribution from scattering and Lyα collisional excitation is negligible, this would imply that the emitting gas should be in the form of dense (n > 1 cm−3), highly ionized and cold (T ∼ 104) structures ("clumps") with volume filling factors smaller than 10−3 or, analogously, with clumping factors larger than about a thousand on projected scales with size of about 5-8 kpc. The apparent ubiquity of giant Lyα nebulae around bright AGN, at least at 3 < z < 4, and in overdense environment suggests that this cold gas should be a common occurrence within and around the halos of massive galaxies. Deep observations of non-resonant lines such He II 1640 and hydrogen Hα from giant Lyα nebulae can better constrain the scattering and collisional contribution to the emission but their are currently lacking for the majority of the nebulae. Ongoing surveys (e.g., with MUSE, MOSFIRE, KMOS and JWST in the future) will soon provide these data and therefore potentially help in refining our understanding of the physical properties of this cold gas.

Lyα integral-field and long-list spectroscopy shows that radio-loud nebulae (i.e. associated with radio-galaxies and radio-loud quasars) are almost invariably associated with larger velocity widths with respect to the majority of radio-quiet systems (see section 4). Together with the analysis of the He II/Lyα and C IV/Lyα ratios, these observations seem to suggest that kinematics in radio-loud nebulae may be dominated by ionized outflows of relatively cold and metal-enriched material within at least the inner 30-50 kpc from the AGN. On the other hand, these observations suggest that ionized and clumpy gas in radio-quiet nebulae should be in a more "stationary" situation and in only in some cases there are possible evidences that this gas is settled in rotating structures. Definitive evidences for the accretion of this cold gas into galaxies from Lyα emission are not clearly detected in the current data. However, it is important to notice that accretion signatures could be "washed-out" by Lyα radiative transfer effects or too small to be detected with current facilities because of their magnitude and possible projection effects. Again, future deep surveys using other bright, non-resonant lines such as hydrogen Hα or He II 1640 would be extremely helpful to search for small velocity shears and therefore for clearer signature of cosmological gas accretion onto galaxies and AGN.

Acknowledgements I gratefully acknowledge support from Swiss National Foundation grant PP00P2_163824.

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