Kolmogorov’s theory of non-helical turbulence is reviewed, and the energy cascade is described, together with the effects of intermittency of the rate of energy dissipation; effects of helicity are then considered, taking account of the invariance of helicity as well as energy in the cascade process. Realisability conditions are obtained. Numerical results use the eddy-damped quasi-Markovian closure scheme (EDQNM). Nonlinear interaction of Alfvén waves leading to the Kraichnan–Iroshnikov spectrum is described. Batchelor’s early analogy between magnetic field and vorticity leading to a dynamo criterion in terms of the magnetic Prandtl number is reviewed. The Malkus–Proctor theory, whereby dynamo saturation is achieved through modification of the mean velocity field, is presented, in some cases leading to a Taylor state with superposed torsional oscillations. Magnetostrophic turbulence, i.e. turbulence in a strongly rotating fluid and in the presence of a strong dynamo-generated magnetic field, is described; here, the helicity that is responsible for dynamo action is generated by the rise of buoyant parcels of fluid driven by either thermal or compositional convection.