Rapid-quench scanning tunnelling microscopy (RQ-STM) has been used to study the growth of InAs on GaAs(001). Particular emphasis will be given to the development of 2D and 3D surface structures for deposition thicknesses close to the critical coverage, since it is at this stage that the number density of quantum dots (QDs) is essentially determined. The wetting layer evolves towards a common alloyed structure prior to 3D growth independently of the starting GaAs reconstruction [either (2´ 4) or c(4´ 4) depending on the starting As pressure]. Comparison will also be made to other low index crystal faces for both the homoepitaxial and heteroepitaxial growth of InAs and GaAs.