An inventory of the baryons at the present epoch shows that the visible galaxies and the hot gas in clusters and rich groups comprise only a fraction of the baryons inferred from Big Bang nucleosynthesis. Most of the baryons must be in a form that is difficult to detect, and cosmological simulations suggest that it is still in the gaseous phase in regions of low and moderate overdensities with temperatures of 1E5 - 1E7 K; this is the \x{201C}missing baryon problem\x{201D}. We have used a few different approaches to detect the presence of this elusive hot medium, such as the shadowing of the cosmological component of diffuse X-ray emission by HI in edge-on galaxies, with a possible detection. We also have sought to detect the presence of baryonic filaments between galaxy clusters (the cosmic web) and find absorption lines associated with these putative filaments. Finally, we studied the X-ray absorption in the Local Group, which some have suggested indicates a gaseous mass significantly greater than the mass of the galaxies; this would help solve the missing baryon problem. However, our work suggests this should be interpreted as a Galactic halo of much lower mass.