  Modeling galaxy formation in a cosmological context presents one of the
greatest challenges in astrophysics today, due to the vast range of scales and
numerous physical processes involved. Here we review the current status of
models that employ two leading techniques to simulate the physics of galaxy
formation: semi-analytic models and numerical hydrodynamic simulations. We
focus on a set of observational targets that describe the evolution of the
global and structural properties of galaxies from roughly Cosmic High Noon
($z\sim 2-3$) to the present. Although minor discrepancies remain, overall,
models show remarkable convergence between different methods and make
predictions that are in qualitative agreement with observations. Modelers seem
to have converged on a core set of physical processes that are critical for
shaping galaxy properties. This core set includes cosmological accretion,
strong stellar-driven winds that are more efficient at low masses, black hole
feedback that preferentially suppresses star formation at high masses, and
structural and morphological evolution through merging and environmental
processes. However, all cosmological models currently adopt phenomenological
implementations of many of these core processes, which must be tuned to
observations. Many details of how these diverse processes interact within a
hierarchical structure formation setting remain poorly understood. Emerging
multi-scale simulations are helping to bridge the gap between stellar and
cosmological scales, placing models on a firmer, more physically grounded
footing. Concurrently, upcoming telescope facilities will provide new
challenges and constraints for models, particularly by directly constraining
inflows and outflows through observations of gas in and around galaxies.
