Galaxies are not spread uniformly through space, but are highly clustered in a network of super-clusters separated by voids, now called the Cosmic Web. The galaxies contain only about 10% of the mass in the Web; the remainder of the baryonic (ordinary) matter is in the highly-ionized intergalactic medium (IGM). The proposing team will conduct observational studies of the IGM, using the Cosmic Origins Spectrograph (COS) aboard the Hubble Space Telescope to probe the gas in absorption in the spectra of quasi-stellar objects (QSOs), and large ground-based telescopes to study the galaxies near those lines of sight with the intent of determining the connection between the galaxies and the absorption lines. They will exploit a database of ~140 QSO sight lines with high-quality COS spectra, and obtain spectra of the galaxies nearby with a combination of archival data from the Sloan Digital Sky Survey (SDSS), new observations with the HYDRA spectrographs at the WIYN and CTIO 4-meter telescope, and a new redshift survey conducted with the Gemini telescopes. The goal of these combined observations is to produce (1) the most complete census of baryons in the Cosmic Web to date, including the hotter gas currently regarded as "missing"; (2) a potential solution to the baryon deficiency problem in spiral galaxies; and (3) a mapping of circumgalactic gas, representing a largely undetected phase of the cycle of infall and outflow that regulates star formation in galaxies. The team will disseminate images, spectra, and higher-level data products via the National Virtual Observatory; train a postdoc and graduate students in optical and UV spectroscopy techniques; adapt these techniques to undergraduate-level exercises; and develop a public planetarium show.