Carbon nanotubes possess unique properties that make them potentially ideal materials for various technological applications. However, a basic growth mechanism explaining the way metallic atoms interact with carbon to nucleate, grow and heal CNTs still needs to be understood. In this review paper we describe the mechanisms of catalytic chemical vapor deposition growth of multi-walled carbon nanotubes and carbon nanofibers, the role of various parameters that govern their growth kinetics and the knowledge added to singlewalled nanotube growth. We also examine future strategies needed to reveal complete knowledge of the growth mechanisms of carbon nanotubes.