The low-redshift Ly-alpha forest of absorption lines provides a probe of large-scale baryonic structures in the intergalactic medium, some of which may be remnants of physical conditions set up during the epoch of galaxy formation. We discuss our recent Hubble Space Telescope (HST) observations and interpretation of low-z Ly-alpha clouds toward nearby Seyferts and QSOs, including their frequency, space density, estimated mass, association with galaxies, and contribution to Omega-baryon. Our HST/GHRS detections of 70 Ly-alpha absorbers with N_HI > 10^12.6 cm-2 along 11 sightlines covering pathlength Delta(cz) = 114,000 km/s show f(>N_HI) ~ N_HI^{-0.63 +- 0.04} and a line frequency dN/dz = 200 +- 40 for N_HI > 10^12.6 cm-2 (one every 1500 km/s of redshift). A group of strong absorbers toward PKS 2155-304 may be associated with gas (400-800) h_75^-1 kpc from 4 large galaxies, with low metallicity (< 0.003 solar) and D/H < 2 x 10^-4. At low-z, we derive a metagalactic ionizing radiation field from AGN of J_0 = 1.3^{+0.8 -0.5} x 10^-23 ergs/cm2/s/Hz/sr and a Ly-alpha-forest baryon density Omega-baryon = (0.008 +- 0.004) h_75^-1 [J_-23 N_14 b_100]^{1/2} For clouds of characteristic size b = (100 kpc)b_100.
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