Mean-field equations are derived for axisymmetric systems, including both the α-effect and the ‘ω-effect’ of differential rotation. The distinction between α2-dynamos and αω-dynamos is explained. The α2-dynamo is treated, first as regards its free modes, then for the space-periodic ‘ABC-flow’, then for a spherical domain of fluid. Numerical results for α2-dynamos when α is antisymmetric about the equatorial plane are presented. The αω-dynamo is then treated, first as regards free modes, in this case taking the form of ‘dynamo waves’, then through a succession of models involving well-separated concentrated layers of α-effect and shear. A model for the galactic dynamo is analysed, showing dynamo modes of dipole or quadrupole symmetry, steady or oscillatory in each case. A ‘stasis’ dynamo of αω-type is described, and a range of numerical results for prescribed distributions of α and ω is presented. The Karlsruhe experiment modelling a space-periodic α2-dynamo and the `VKS’ experiment modelling an αω-dynamo with or without field reversals are described. Finally, the evolving Taylor–Green vortex and the manner in which concentrated flux tubes emerge from the dynamo action associated with this flow are presented.