Over the past quarter-century, there has been intensive research into electron transport in low dimensional systems, especially those fabricated in semiconductor material. This research has added to our understanding of electrical transport properties outside of the macroscopic regime, and allowed exploration of potential device designs and fabrication methods. To define devices on micron and nanometer length scales, powerful lithographic techniques such as photolithography and electron-beam lithography have been widely employed to pattern semiconductor material. A newer technique is exploited in this research, where an atomic force microscope (AFM) performs local anodic oxidation on the surface of a shallow GaAs/ AlGaAs heterostructure. At cryogenic temperatures the resulting oxide lines define in-plane electrostatic gates of submicron size, and one-dimensional ballistic channels exhibiting quantized conduction. These techniques are also used to fabricate T-shaped quantum mechanical transistors. The conductance through these devices can be modulated by adjusting the voltage on a nearby in-plane gate, which changes the geometry of the transistor. This in turn shifts the energy of a bound electron state within the channel, causing reflection of the conduction electrons when this state is brought into resonance with the channel. For optimum operation, this device requires a hard-wall potential profile, which is a benefit of the AFM local anodic oxidation technique. The AFM oxidation technique is also used to pattern wafers of GaAs which are later cleaned and regrown in a molecular beam epitaxy system. GaAs and InAs are deposited on top of the patterned semiconductor material; the InAs self-assembles into quantum dots with zero-dimensional electron and hole states. These quantum dots, if buried under subsequent GaAs growth, are optically active and potentially useful in a wide variety of optoelectronic devices. The presence of the oxide patterns is shown to locally vary the dot density, and by careful calibration of oxide lithography and growth parameters, site selection of single dots or small populations of InAs quantum dots can be achieved.