A major obstacle for the applications of single-walled carbon nanotubes (SWNTs) in electronic devices is their structural diversity, ending in SWNTs with diverse electrical properties. Catalytic chemical vapor deposition has shown great promise in directly synthesizing high-quality SWNTs with a high selectivity to specific chirality (n, m). During the growth process, the tube-catalyst interface plays crucial roles in regulating the SWNT nucleation thermodynamics and growth kinetics, ultimately governing the SWNT chirality distribution. Starting with the introduction of SWNT growth modes, this review seeks to extend the knowledge about chirality-selective synthesis by clarifying the energetically favored SWNT cap nucleation and the threshold step for SWNT growth, which describes how the tube-catalyst interface affects both the nucleus energy and the new carbon atom incorporation. Such understandings are subsequently applied to interpret the (n, m) specific growth achieved on a variety of templates, such as SWNT segments or predefined molecular seeds, transition metal (Fe, Co and Ni)-containing catalysts at low reaction temperatures, W-based alloy catalysts, and metal carbides at relatively high reaction temperatures. The up to date achievements on chirality-controlled synthesis of SWNTs is summarized and the remaining major challenges existing in the SWNT synthesis field are discussed.