Low temperature electrical measurements on particular semiconductor structures can exhibit effects due to the spatial confinement of carriers and reduced dimensionality. Examples are provided by studies of tunnelling in confining structures, transport in narrow channels, ballistic transport through constrictions and quantum interference in rings. These observations rely on the growth of layers by molecular beam epitaxy (MBE) to provide both two-dimensional electron gases (2DEGs) and the very high material quality essential for the observation of some quantum effects. To define the specific geometries necessary for the above experiments electrostatic "squeezing" can be used to confine the planar carrier gas to regions of controllable extent. Recent results obtained from MBE GaAs/AlGaAs 2DEG structures can now be compared with measurements made on hole gases and the use of in situ ion beams in fabrication promises a new class of structures for investigation. However, it is rarely sufficient merely for MBE to provide a carrier gas with the highest maximum carrier mobility. The particular requirements of an experiment affect not only the design of the structure itself but also the MBE growth strategy and the optimum deposition conditions. Growth rate, substrate temperature and source material flux ratios are MBE variables and the suitability of layers for specific postgrowth physics experiments can be shown to depend upon these variables in a systematic manner. Moreover, further requirements for the grown material can include the need to provide specific sheet carrier concentrations, high mobilities in unilluminated samples and the complete absence of conducting paths in parallel with the 2DEG. There are often compromises inherent in attempting to simultaneously optimise several MBE variables and careful management of the MBE machine itself is essential. © 1989 IOP Publishing Ltd.