| Only in recent years the realization has emerged that galaxies do not dominate the universal baryon budget but are merely the brightest pearls of an underlying Cosmic Web. Although the gas in these intergalactic filaments is moderately to highly ionized, QSO absorption lines have shown that the surface area increases dramatically in going down to lower HI column densities. The first image of the Cosmic Web in HI emission has just been made of the Local Group filament connecting M31 and M33. The corresponding HI distribution function is in very good agreement with that of the QSO absorption lines, confirming the 30-fold increase in surface area expected between 10^19 cm^-2 and 10^17 cm^-2. The critical observational challenge is crossing the "HI desert", the range of log(N_HI) from about 19.5 down to 18, over which photo-ionization by the intergalactic radiation field produces an exponential decline in the neutral fraction from essentially unity down to a few percent. Nature is kinder again to the HI observer below log(N_HI)=18, where the neutral fraction decreases only very slowly with log(N_HI). We have defined a research program comprising two complementary initiatives which should lay the basis for a comprehensive study of the Cosmic Web phenomenon. The first is a complete reprocessing and analysis of the recently published HI Parkes All-Sky Survey (HIPASS) of the HI sky between declinations of -90 and +20 degree after optimization for the detection of diffuse filaments near bright galaxies. The second initiative involves a novel and extremely sensitive method for detection of faint HI emission features which employs the WSRT to simulate a filled aperture, while retaining the excellent spectral baseline and PSF properties of an interferometer. The observed HI linewidth of detected features should permit a clear assessment of the relative importance of "cold-mode" versus "hot-mode" accretion, and its dependence on environment; providing important constraints on this critical open issue in cosmology and galaxy evolution. |