We study the formation of damped \lya and Lyman limit absorbers in a hierarchical clustering scenario using a gas dynamical simulation of an $\Omega = 1$, cold dark matter universe. In the simulation, these high column density systems are associated with forming galaxies. Damped \lya absorption, $N_{HI} \simgt 10^{20.2}\cm^{-2}$, arises along lines of sight that pass near the centers of relatively massive, dense protogalaxies. Lyman limit absorption, , develops on lines of sight that pass through the outer parts of such objects or near the centers of smaller protogalaxies. The number of Lyman limit systems is less than observed, while the number of damped \lya systems is quite close to the observed abundance. Damped absorbers are typically kpc in radius, but the population has a large total cross section because the systems are much more numerous than present day galaxies. Our results demonstrate that high column density systems like those observed arise naturally in a hierarchical theory of galaxy formation and that it is now possible to study these absorbers directly from numerical simulations.

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