The most important practical characteristic of a superconductor is its critical current density. This article traces the history, both of the experimental discoveries and of the development of the theoretical ideas that have lead to the understanding of those factors that control critical current densities. These include Silsbee`s hypothesis, the Meissner effect, London, Ginsburg - Landau and Abrikosov theories, flux pinning and the critical state, and the control of texture in high temperature superconductors.