We review theoretical approaches to the study of galaxy formation, with emphasis on the role of hydrodynamic simulations in modeling the high redshift galaxy population. We present new predictions for the abundance of star-forming galaxies in the Lambda + cold dark matter model (Omega_m=0.4, Omega_L=0.6), combining results from several simulations to probe a wide range of redshift. At a threshold density of one object per arcmin^2 per unit z, these simulations predict galaxies with star formation rates of 2 msun/yr (z=10), 5 msun/yr (z=8), 20 msun/yr (z=6), 70-100 msun/yr (z=4-2), and 30 msun/yr (z=0.5). For galaxies selected at a fixed comoving space density n=0.003 h^3 Mpc^{-3], a (50 Mpc/h)^3 simulation predicts a galaxy correlation function (r/5 Mpc/h)^{-1.8} in comoving coordinates, essentially independent of redshift from z=4 to z=0.5. Different cosmological models predict global histories of star formation that reflect their overall histories of mass clustering, but robust numerical predictions of the comoving space density of star formation are difficult because the simulations miss the contribution from galaxies below their resolution limit. The LCDM model appears to predict a star formation history with roughly the shape inferred from observations, but it produces too many stars at low redshift, predicting Omega_* ~ 0.015 at z=0. We conclude with a brief discussion of this discrepancy and three others that suggest gaps in our current theory of galaxy formation: small disks, steep central halo profiles, and an excess of low mass dark halos. While these problems could fade as the simulations or observations improve, they could also guide us towards a new understanding of galactic scale star formation, the spectrum of primordial fluctuations, or the nature of dark matter.
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