An introductory review of the central ideas in the kinetics of submonolayer and multilayer expitaxial growth is followed by a more detailed discussion of some recent developments in the field. The concepts of a critical island size, dynamical scaling of the island-size distribution, and the barrier to interlayer diffusion (Ehrlich-Schwoebel barrier) are introduced. The results of kinetic Monte Carlo simulations of a realistic model of submonolayer epitaxial growth are presented and compared with rate-equation analyses and recent experiments. We also present an analytical expression for the scaled island-size distribution as a function of the critical island size which agrees well with our simulations as well as with experiments. Our results provide a quantitative explanation for the variation of the submonolayer island density, critical island size, island-size distribution and morphology as a function of temperature and deposition rate found in recent experiments. We also present the results of a realistic model for multilayer growth which includes a finite barrier to interlayer diffusion. A method for determining the Ehrlich-Schwoebel barrier based on a comparison of simulations with experimental results for the reflection high-energy electron diffraction intensity, surface width, layer densities, and surface morphology is discussed. In particular, we find that for Fe/Fe(100) the interlayer diffusion barrier is significantly less than the activation energy for diffusion on a flat terrace.