We present atomic-scale, video-rate environmental transmission electron microscopy and in situ time-resolved X-ray photoelectron spectroscopy of surface-bound catalytic chemical vapor deposition of single-walled carbon nanotubes and nanofibers. We observe that transition metal catalyst nanoparticles on SiOx support show crystalline lattice fringe contrast and high deformability before and during nanotube formation. A single-walled carbon nanotube nucleates by lift-off of a carbon cap. Cap stabilization and nanotube growth involve the dynamic reshaping of the catalyst nanocrystal itself. For a carbon nanofiber, the graphene layer stacking is determined by the successive elongation and contraction of the catalyst nanoparticle at its tip.