We have used a variety of computational methods to study key aspects of single-walled carbon nanotube (SWNT) growth. Molecular dynamics (MD) studies based on an empirical force field showed; for example; why SWNT growth occurs in a temperature window and why; for 1-2 nm catalyst particles; the SWNT diameter varies linearly with the size of the particle. In addition; the liquid or solid phase of the catalyst particle is strongly dependent on particle size; and smaller particles (< 1.5 nm) are liquid at typical chemical vapor deposition temperatures whereas larger particles (> 5 nm) are solid. The phase of particles of intermediate sizes depends on the exact temperature and on their carbon content. The effect of substrates on metal-carbide properties and SWNT growth has been studied by combing density functional (DFT) and MD methods. A major effect of flat; inert substrates is to flatten the catalyst particles thereby increasing their melting points. DFT has also been used to study the catalyst-SWNT interaction which is critical for the growth of long SWNTs; and is also being used to study the importance of the SWNT cap structure on its chirality. This knowledge is important; for example; when using SWNTs as seeds for the growth of longer nanotubes.