The striking flow shown in figure 1 is produced in a simple apparatus: A fluid is confined between two concentric cylinders, with the inner and perhaps the outer cylinder able to rotate. The cellular motion that develops with rotation was discovered and described mathematically by Geoffrey I. Taylor in 1923. A similar apparatus, with the inner cylinder suspended from a torsion fiber and the outer cylinder rotating, was used even earlier as a viscometer. Maurice Couette described this arrangement in his thesis, which he presented in Paris in 1890. For this reason, modern investigators refer to flow between rotating cylinders as Taylor‐Couette flow. In this article I trace the beginnings of the subject back to Isaac Newton and, by discussing the contributions of Newton, George Stokes, Max Margules, Arnulph Mallock, Couette, Taylor, S. Chandrasekhar and others, show how the study of this flow evolved to its place of prominence today.