Single-wall carbon nanotubes (SWCNTs) with a unique tubular structure have exhibited excellent electrical, thermal and mechanical properties. However, their attractive applications in microelectronic devices and sensors are still pending due to the lack of high-quality, structure-defined SWCNTs. It is still a great challenge to grow pure and ordered SWCNTs with designated structures and properties. The key of breakthrough is to understand the fundamental nucleation and growth mechanism of SWCNTs under reaction conditions. In this article, we analyze the influences of physical and chemical properties, such as electronic structure, melting point, carbon solubility, and diffusivity, of catalyst nanoparticles on the productivity, purity, and fine structures of grown SWCNTs. The progress, current situation, and challenges on the controlled growth of SWCNTs are summarized. Finally, perspectives on future directions are presented and a strategy of structure-controlled production of SWCNTs is proposed.