In this paper we discuss our recent results on the homoepitaxial growth of GaAs from beams of Ga and As 2( 4) and of InAs on GaAs from In and As 2( 4) beams. Experimental measurements are based on in-situ reflection high energy electron diffraction (RHEED) and scanning tunnelling microscopy (STM), while their analysis relies on kinetic Monte Carlo (KMC) simulations and the application of rate equations. We emphasise the comparative behaviour on all three low-index substrate orientations and the importance of surface reconstruction. Wherever possible we use an atomistic approach and in the homoepitaxial studies we are principally concerned with nucleation effects occurring at the sub-monolayer stage, including site-specific adatom incorporation and arsenic molecule dissociation pathways, especially the essential requirement of a mobile precursor state. Some results for vicinal plane growth are also presented. In the case of the InAs system, we have investigated strain relaxation processes using STM images, while specifically for growth on GaAs(001)-2×4 and c(4×4) surfaces we have studied the formation of so-called self assembled quantum dots (SADs) in relation to the classical Stranski–Krastanov mechanism and the associated strain effects. Although it is frequently stated that this accounts fully for dot formation, we have found many anomalies and believe the process, and indeed the final structures, to be much more complex than generally believed.