Carbon nanotubes have been subject to a great many theoretical and experimental investigations and have many interesting properties. However, the caps found at the ends of nanotubes have been rather neglected. Previous work has established the possible caps for a given nanotube. This thesis seeks to build on this work to determine which caps are probable. Three representative nanotubes are considered: the (5,5) tube (an example of a metal); the (10,0) tube (an example of an insulator) and the (11,2) tube (for which experimental data is available). A linear scaling density matrix method based on orthogonal tight-binding theory is used for a systematic study of doubly capped nanotubes. The energetically most stable caps are found for all three tubes and the isolated pentagon rule established for fullerenes is shown to be valid for nanotube caps. No simple rule governing the stability of isolated pentagon caps is found, although the stability of (11,2) caps correlates well with the number of hexagons adjacent to just one pentagon. The local densities of states (LDOS) are calculated for the most stable cap for each nanotube. A localised state is found for the capped (5,5) tube and a resonant state is observed for the capped (11,2) tube. The LDOS for the capped (11,2) tube is compared with experimental observations and questions are raised concerning the nature of the tube observed and the limitations of STM as a method for identifying nanotube caps.