Since their discovery in the early 1990s, single-walled carbon nanotubes (SWNTs) have spawned previously unimaginable commercial and industrial technologies. Their versatility stems from their unique electronic, physical/chemical, and mechanical properties, which set them apart from traditional materials. Many researchers have investigated SWNT growth mechanisms in the years since their discovery. The most prevalent of these is the vapor-liquid-solid (VLS) mechanism, which is based on experimental observations. Within the VLS mechanism, researchers assume that the formation of a SWNT starts with co-condensation of carbon and metal atoms from vapor to form liquid metal carbide. Once the liquid reaches supersaturation, the solid phase nanotubes begin to grow. The growth process is partitioned into three distinct stages: nucleation of a carbon "cap-precursor," "cap-to-tube" transformation, and continued SWNT growth. In recent years, molecular dynamics (MD) simulations have come to the fore with respect to SWNT growth. MD simulations lead to spatial and temporal resolutions of these processes that are superior to those possible using current experimental techniques, and so provide valuable information regarding the growth process that researchers cannot obtain experimentally. In this Account, we review our own recent efforts to simulate SWNT nucleation, growth, and healing phenomena on transition-metal catalysts using quantum mechanical molecular dynamics (QM/MD) methods. In particular, we have validated each stage of the SWNT condensation mechanism using a self-consistent-charge density-functional tight-binding (SCC-DFTB) methodology. With respect to the nucleation of a SWNT cap-precursor (stage 1), we have shown that the presence of a transition-metal carbide particle is not a necessary prerequisite for SWNT nucleation, contrary to conventional experimental presumptions. The formation and coalescence of polyyne chains on the metal surface occur first, followed by the formation of the SWNT cap-precursor, "ring condensation", and the creation of an sp(2)-hybridized carbon structure. In our simulations, the nucleation process takes approximately 400 ps. This first step occurs over a much longer time scale than the second stage of SWNT condensation (approximately 50 ps). We therefore observe SWNT nucleation to be akin to the rate-limiting step of the SWNT formation process. In addition to the QM/MD simulation of various stages of SWNT nucleation, growth, and healing processes, we have determined the effects of temperature, catalyst composition, and catalyst size on the kinetics and mechanism of SWNT growth. With respect to temperature dependence, we observe a "sweet-spot" with respect to the efficiency of SWNT growth. In addition, Ni-catalyzed SWNT growth is observed to be 70-100% faster compared to Fe-catalyzed SWNT growth, depending on the catalyst particle size. We also observe a noticeable increase in SWNT growth rates using smaller catalyst particles. Finally, we review our recent QM/MD investigation of SWNT healing. In particular, we recount mechanisms by which adatom defects, monovacancy defects, and a "5-7 defect" are removed from a nascent SWNT. The effectiveness of these healing mechanisms depends on the rate at which carbon moieties are incorporated into the growing SWNT. Explicitly, we observe that healing is promoted using a slower carbon supply rate. From this rudimentary control of SWNT healing, we propose a route towards chirality-controlled SWNT growth.