With close pairs of quasars at different redshifts, a background quasar sightline can be used to study a foreground quasar's environment in absorption. We use a sample of 650 projected quasar pairs to study the H I Lyα absorption transverse to luminous, z ∼ 2 quasars at proper separations of 30 kpc < R⊥< 1 Mpc. In contrast to measurements along the line-of-sight, regions transverse to quasars exhibit enhanced H I Lyα absorption and a larger variance than the ambient intergalactic medium, with increasing absorption and variance toward smaller scales. Analysis of composite spectra reveals excess absorption characterized by a Lyα equivalent width profile W = 2.3 Å (R⊥/100 kpc)-0.46. We also observe a high (≃ 60%) covering factor of strong, optically thick H I absorbers (H I column ) at separations R⊥< 200 kpc, which decreases to ∼20% at R⊥≃ 1 Mpc, but still represents a significant excess over the cosmic average. This excess of optically thick absorption can be described by a quasar-absorber cross-correlation function ξQA(r) = (r/r0)γwith a large correlation length (comoving) and . The H I absorption measured around quasars exceeds that of any previously studied population, consistent with quasars being hosted by massive dark matter halos Mhalo≈ 1012.5M⊙at z ∼ 2.5. The environments of these massive halos are highly biased toward producing optically thick gas, and may even dominate the cosmic abundance of Lyman limit systems and hence the intergalactic opacity to ionizing photons at z ∼ 2.5. The anisotropic absorption around quasars implies the transverse direction is much less likely to be illuminated by ionizing radiation than the line-of-sight. © 2013. The American Astronomical Society. All rights reserved..