This article outlines the early development of the undulator in which an electron beam travelling through a system of magnetic fields with directions alternating in space emits a species of linearly polarized synchrotron radiation peaked at a wavelength which is short compared to the magnet period when the electron energy is high compared with the rest energy. The conditions of synchronism and the departure from synchronism that leads to gain during a single passage of the beam are examined. In later developments, stemming from this, single pass gain of circularly polarized radiation was experimentally demonstrated by means of a ‘twister’ structure and an optical cavity, obtained by adding mirrors to the magnet assembly, was used to achieve build-up of the radiation in an oscillator device known as the ‘free-electron laser’. Applications of this device and further theoretical developments are discussed.