Electron transport in Si inversion layers at 300 K is studied using a self-consistent Monte Carlo solution of the Boltzmann transport equation coupled to the two-dimensional Poisson equation and the one-dimensional Schr\"odinger equation. Physical elements included in the model are (1) nonparabolicity effects to treat quantization in the inversion layer; (2) static screening of the Coulomb interactions accounting for the population of many subbands; (3) anisotropy of the deformation-potential interaction, shown to be quite important in the case of a two-dimensional electron gas (2DEG); (4) a careful analysis of the dynamic screening of the deformation-potential interaction, showing that the interaction between electrons and acoustic phonons can be approximated by the unscreened interaction in the nondegenerate limit of a 2DEG; and (5) the inclusion of interface ${\mathrm{SiO}}_{2}$ optical phonons. Up to ten subbands have been included to study the 2DEG together with a bulk-transport model employed to handle high-energy electrons. We have obtained mixed results: In the Ohmic regime, we have found a phonon-limited mobility that exhibits the correct dependence on carrier density, but which is about 20% larger than the experimental data. This still represents an improvement upon previous nonempirical theories, and even better quantitative agreement is obtained at the very low and very high carrier densities at which Coulomb scattering and scattering with surface roughness, respectively, control the mobility. At high longitudinal fields we find a bulklike saturated velocity, in agreement with some experimental results, but not with many others that we consider more reliable.