Amyloid fibrils are β-sheet-rich linear protein polymers that can be formed by a large variety of different proteins. These assemblies have received much interest in recent decades, due to their role in a range of human disorders. However, amyloid fibrils are also found in a functional context, whereby their structural, mechanical and thermodynamic properties are exploited by biological systems. Amyloid fibrils form through a nucleated polymerisation mechanism with secondary processes acting in many cases to amplify the number of fibrils. The filamentous nature of amyloid fibrils implies that the fibril growth rate is, by several orders of magnitude, the fastest step of the overall aggregation reaction. This article focusses specifically on in vitro experimental studies of the process of amyloid fibril growth, or elongation, and summarises the state of knowledge of its kinetics and mechanisms. This work attempts to provide the most comprehensive summary, to date, of the available experimental data on amyloid fibril elongation rate constants and the temperature and concentration dependence of amyloid fibril elongation rates. These data are compared with those from other types of protein polymers. This comparison with data from other polymerising proteins is interesting and relevant because many of the basic ideas and concepts discussed here were first introduced for non-amyloid protein polymers, most notably by the Japanese school of Oosawa and co-workers for cytoskeletal filaments.