Chandra and FUSE view of the WHIM: the Local Group and beyond

In this contribution, I review the current observational evidence for the existence of filaments of Warm-Hot Intergalactic Medium (WHIM). In particular, I first focus on the controversial issue of the identification of the $z\sim 0$ highly ionized far-ultraviolet (i.e. OVI) and X-ray (i.e. OVII, OVIII and NeIX) absorbers with either a very tenuous and diffuse WHIM filament, or with much denser condensations of material at large distances in the Galactic halo. I then present our recent detection (confidence level $>3\sigma$) of the OVII WHIM at $z>0$ and derive an estimate of the total number of baryons contained in this hard-to-detect phase of the IGM.