Dynamo theory is introduced as the theory of the spontaneous generation of magnetic fields by internal inductive motions in planets, stars and galaxies. The historical background to dynamo theory is described, focussing mainly on the magnetic fields of the Earth and the Sun. The simplest self-exciting dynamo model (the homopolar disc dynamo) is described, and its limitations are indicated. Cowling’s anti-dynamo theorem is touched on, and the resulting need for departures from axisymmetry in the internal fluid motions is discussed. For flows that are turbulent (whether strongly nonlinear or weak, as in a field of random waves), the `mean-field’ approach is described, and the need for a lack of reflectional symmetry (or ‘chirality’) in the flow is stressed. Other highlights in the historical development since the 1950s are described: the analogy with vorticity in turbulence, the first rigorous examples of dynamo action in a simply connected fluid domain (the two-sphere model of Herzenberg and the stasis model of Backus), the early computational attack on the geodynamo problem by Bullard and Gellman, the `cyclonic events’ theory of E. N. Parker, and the incorporation of dynamic constraints imposed by the Navier–Stokes equation in a strongly rotating fluid.