Due to their extraordinary electronic properties, Quantum Dots (QDs) are potentially very useful nanoscale devices and research tools. As their electrons are confined in all three dimensions, the energy spectra of QDs is descrete, similar to atoms and molecules. Because the gaps between these energy levels is inversely related to the size of the QD, very small QDs are desirable. Carbon nanotubes have long been touted as fundamental units of nanotechnology, due to their structural, optical and electronic properties, many of which are a result of the confinement of electrons in the trans-axial plane of the nanotube. It is known that their band gap structure is altered under deformation of their cross section. It is proposed that one way to fabricate a very small quantum dot is by confining electrons in the nanotube so that they may not freely move along its length. A structure to produce this confinement has been described elsewhere, namely the carbon nanotube cross, consisting of two carbon nanotubes, with the the one draped over the other at ninety degrees. It is thought that this structure will induce local physical deformations in the nanotube, resulting in local changes in electronic structure of the top nanotube at the junction of the cross. These band gap shifts may cause metal-semiconductor transitions, resulting in tunnel barriers that axially the confine electrons in the nanotube. This thesis investigates the possibility that the carbon nanotube cross may exhibit QD behavior at the junction of the cross, due to these local band gap shifts. A device for carbon nanotube growth, using Chemical Vapor Deposition, has been designed, and may be built using microfabrication techniques. This device consists of electrodes (for electrical measurements of the nanotubes) and catalyst regions (to initiate nanotube growth), lithographically patterned in a configuration that promotes carbon nanotube formation. Unfortunately, due to fabrication issues, this effort is a work in progress, and these devices have not yet been constructed. However, an experimental methodolgy has been developed, which provides a framework for eventually building a carbon nanotube cross, and investigating the possibility of QD behavior at the junction of the cross. This structure has also been investigated computationally. Molecular dynamics simulations were used to obtain equilibrium geometries of the carbon nanotube cross, and it was found that their are many different meta stable states, corresponding to different types of nanotube, and different physical arrangements of these nanotubes. The electronic structure of the carbon nanotube cross was calculated using the density functional theory. Band gap energies similar to experimental values were obtained. A one-to-one spatial correlation between deformation and band gap and conduction band shifts were observed in the top carbon nanotube of the nanotube cross. Small tunnel barriers, inferred from both the calculated band gap and LUMO energies, are observed, and could well be sufficient to confine electrons along the axis of the nanotube. The results described in this thesis, while not definitive, certainly indicate that a QD probably would form at the junction of a carbon nanotube cross, and that further investigation, both experimental and computational, is warranted.