A review is given of nucleation and growth models as applied to the earliest stages of thin film growth. Rate equations, kinetic Monte Carlo, and level set: simulations are described in some detail, with discussion of remaining uncertainties, in particular the functional form of the so-called capture numbers in rate equations. Recent examples are given of sub-monolayer nucleation at surface defects, attachment-limited capture, and Ostwald ripening: The experimental literature is cited, and experiment-theory comparisons are made where possible. Emphasis is given to fast computational models that can span a large range of length and time scales, which might be further developed in the direction of on-line process control. (C) 2003 American Vacuum Society.