Undoubtedly, the catalyst plays a key role in the nucleation and growth of various carbon filaments, as evidenced by the large number of studies for more than 40 years that focus on catalyst preparation methods, composition, morphology and size. Variation of catalyst compositions have led to morphological control of grown carbon filaments such as vermicular, twisted, platelet, tubular and conical. Reduction of the catalyst nanoparticle size to the 1–5 nm range resulted in the formation of new allotrope, carbon nanotubes, with unique properties that depend on their structure. It is commonly accepted that a careful designed catalyst could control the structure of grown single walled carbon nanotubes and thereby their properties. However, a catalyst a few nanometers in diameter introduces peculiarities for classical understanding of the features e.g., melting temperature, binary/ternary phase diagrams with carbon and other metals, equilibrium shape (Wulff construction), crystallographic structure, impact of the substrate, etc. Current state of the art of thermodynamics and kinetics of nanoparticles for controllable growth of carbon nanotubes and challenges will be presented and discussed.