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We analyze the IllustrisTNG simulations to study the mass, volume fractionand phase distribution of gaseous baryons embedded in the knots, filaments,sheets and voids of the Cosmic Web from redshift $z=8$ to redshift $z=0$. Wefind that filaments host more star-forming gas than knots, and that filamentsalso have a higher relative mass fraction of gas in this phase than knots. Wealso show that the cool, diffuse Intergalactic Medium (IGM; $T<10^5 \, {\rmK}$, $ n_{\rm H}<10^{-4}(1+z) \, {\rm cm^{-3}}$) and the Warm-Hot IntergalacticMedium (WHIM; $ 10^5 \, {\rm K} <T<10^7 \, {\rm K}$, $ n_{\rm H}<10^{-4}(1+z)\, {\rm cm^{-3}}$) constitute $\sim 39\%$ and $\sim 46\%$ of thebaryons at redshift $z=0$, respectively. Our results indicate that the WHIM mayconstitute the largest reservoir of {\it missing} baryons at redshift $z=0$.Using our Cosmic Web classification, we predict the WHIM to be the dominantbaryon mass contribution in filaments and knots at redshift $z=0$, but not insheets and voids where the cool, diffuse IGM dominates. We also characterisethe evolution of WHIM and IGM from redshift $z=4$ to redshift $z=0$, and findthat the mass fraction of WHIM in filaments and knots evolves only by a factor$\sim 2$ from redshift $z=0$ to $z=1$, but declines faster at higher redshift.The WHIM only occupies $4-11\%$ of the volume at redshift $0\leq z \leq 1$. Wepredict the existence of a significant number of currently undetected OVII andNeIX absorption systems in cosmic filaments which could be detected by futureX-ray telescopes like Athena.

Baryons in the CosmicWeb of IllustrisTNG - I: Gas in Knots, Filaments, Sheets and Voids