We analyze the IllustrisTNG simulations to study the mass, volume fraction and phase distribution of gaseous baryons embedded in the knots, filaments, sheets and voids of the Cosmic Web from redshift z = 8 to redshift z = 0. We find that filaments host more starforming gas than knots, and that filaments also have a higher relative mass fraction of gas in this phase than knots. We also show that the cool, diffuse Intergalactic Medium (IGM; T < 10 5 K, n H < 10 −4 (1 + z) cm −3 ) and the Warm-Hot Intergalactic Medium (WHIM; 10 5 K < T < 10 7 K, n H < 10 −4 (1 + z) cm −3 ) constitute ∼ 39% and ∼ 46% of the baryons at redshift z = 0, respectively. Our results indicate that the WHIM may constitute the largest reservoir of missing baryons at redshift z = 0. Using our Cosmic Web classification, we predict the WHIM to be the dominant baryon mass contribution in filaments and knots at redshift z = 0, but not in sheets and voids where the cool, diffuse IGM dominates. We also characterise the evolution of WHIM and IGM from redshift z = 4 to redshift z = 0, and find that the mass fraction of WHIM in filaments and knots evolves only by a factor ∼ 2 from redshift z = 0 to z = 1, but declines faster at higher redshift. The WHIM only occupies 4 − 11% of the volume at redshift 0 z 1. We predict the existence of a significant number of currently undetected OVII and NeIX absorption systems in cosmic filaments which could be detected by future X-ray telescopes like Athena.