The Condensed Matter Theory Group has for many years maintained an interest in semiconductor heterostructures and low-dimensional systems, being concerned with a variety of electronic, optical and electronic transport properties. A key factor in understanding the properties of low-dimensional semiconductor structures is an appropriate description of the electronic states. We have made use of a variety of calculational methods, including simple envelope function, k.p. and empirical pseudopotential complex band structure approaches. First principles methods have also been employed, in particular to investigate the physical and electronic structure of very small semiconductor clusters (a form of quantum dot) and the interfaces between different semiconductors which occur in heterostructures.