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We describe cosmological simulation techniques and their application to studies of cosmic structure formation, with particular attention to recent hydrodynamic simulations of structure in the high redshift universe. Collisionless N-body simulations with Gaussian initial conditions produce a pattern of sheets, filaments, tunnels, and voids that resembles the observed large scale galaxy distribution. Simulations that incorporate gas dynamics and dissipation form dense clumps of cold gas with sizes and masses similar to the luminous parts of galaxies. Models based on inflation and cold dark matter predict a healthy population of high redshift galaxies, including systems with star formation rates of 20 M_{\sun}/year at z=6. At z~3, most of the baryons in these models reside in the low density intergalactic medium, which produces fluctuating Lyman-alpha absorption in the spectra of background quasars. The physical description of this ``Lyman-alpha forest'' is particularly simple if the absorption spectrum is viewed as a 1-dimensional map of a continuous medium instead of a collection of lines. The combination of superb observational data and robust numerical predictions makes the Lyman-alpha forest a promising tool for testing cosmological models.


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