Dynamo theory focuses on the generation of an axisymmetric (or mean) magnetic field by the action of a mean electromotive force (e.m.f.) and differential rotation. A topic that has received somewhat less attention is that of the stability of the field to nonaxisymmetric perturbations. In mean-field dynamo theory, the field is maintained when the generation effect of the mean e.m.f. and differential rotation balance the decay due to ohmic diffusion. However, if the field is sufficiently strong and it satisfies certain other conditions then the field may be unstable. The instability extracts energy from the mean field so the generation mechanism may have a second sink of energy to counteract. Magnetic instabilities may therefore play an important role in determining what fields are observed and how strong they are. Theoretical and observational ideas are now converging. The idea that instability might be the mechanism for initiating a field reversal was suggested some time ago. Linear theory has established that the minimum field strength required for instability (though depending on many factors) is comparable with estimates of the Earth's toroidal field strength. More recently, a careful analysis of the reversal data has concluded that ‘reversals are triggered by internal instabilities of the fluid motion of the core’. Here, we review the various classes of magnetic instability and the conditions required for instability.