Our standard cosmological model predicts that most of the matter in the universe is distributed into a network of filaments - the Cosmic Web - in which galaxies form and evolve. Because most of this material is too diffuse to form stars, its direct imaging has remained elusive for several decades leaving fundamental questions about the structure of the universe still open, including: How are galaxies linked to each other? What are the morphological and physical properties of the Cosmic Web on both large and small scales? How do galaxies accrete gas from the Cosmic Web? In this talk, I will tackle these questions using the results of a new program to directly detect and study high-redshift cosmic gas in emission using bright quasars and galaxies as external "sources of illumination’’. In particular, I will show results from ultra-deep narrow-band imaging and integral-field-spectroscopy with both MUSE/VLT and the Keck Cosmic Web Imager (KCWI) that revealed numerous giant Lyman-alpha emitting filaments extending up to several hundred kpc around quasars and bright galaxies. I will discuss how the unexpectedly high luminosities of these systems, together with the constraints from Helium and metal extended emission, represent a challenge for our current understanding of cosmological structure formation. In particular, I will show that current observations suggest that intergalactic gas around high-redshift galaxies and quasars has a much broader density distribution of cold material than expected from cosmological simulations and I will present our first attempts to understand the origin and nature of these structures using high-resolution hydrodynamical models.