The core accretion—gas capture model is generally accepted as the standard formation model for gas giant planets. It proposes that a solid core grows via the accretion of planetesimals, and then captures a massive envelope from the solar nebula gas. Simulations have been successful in explaining many features of giant planets. This chapter will present an overview of the historical and scientific developments of the model, a description of the computer code based on the core accretion hypothesis with a summary of results of recent computer simulations, and the effect the observational achievement of finding extrasolar planets has had on the core accretion—gas capture model.