We present a brief review of the most important efforts aimed at simulating single-walled carbon nanotube (SWNT) nucleation and growth processes using molecular dynamics (MD) techniques reported in the literature. MD simulations allow the spatio-temporal movement of atoms during nonequilibrium growth to be followed. Thus, it is hoped that a successful MD simulation of the entire SWNT formation process will assist in the design of chirality-specific SWNT synthesis techniques. We give special consideration to the role of the metal catalyst particles assumed in standard theories of SWNT formation, and describe the actual metal behavior observed in the reported MD simulations, including our own recent quantum chemical MD simulations. It is concluded that the use of a quantum potential is essential for a qualitatively correct description of the catalytic behavior of the metal cluster, and that carbide formation does not seem to be a necessary requirement for nucleation and growth of SWNTs according to our most recent quantum chemical MD simulations.