Young neutron stars are convective, and amplification of their magnetic fields seems almost inevitable. Fields as strong as 3 x 10(15) G are generated during a period of entropy-driven convection, if the ratio of mean magnetic pressure to turbulent pressure equals that observed in the upper convection zone of the Sun. The approximately 1 ms convective overturn time is probably much shorter than the initial rotation period of a typical pulsar. The energy for field amplification comes mostly from convection, not differential rotation, if the initial rotation period exceeds approximately 30 ms, and probably also for significantly shorter periods. Thus rotation plays a smaller role than in the solar dynamo, and the dominant mode of field amplification will have a scale much smaller than the stellar radius. Large-scale alpha - OMEGA dynamo action is nonetheless possible in protopulsars with rotation periods P(rot) less than or similar to 30 ms during the last stages of neutrino diffusion, and perhaps earlier if there exists a postbounce phase of slow mixing before the onset of entropy-driven convection. Neutron star convection is a transient phenomenon and has an extremely high magnetic Reynolds number, R(m) approximately 10(17). In this sense, a neutron star dynamo is the quintessential fast dynamo. The convective motions are only mildly turbulent on scales larger than the approximately 10(2) cm neutrino mean free path, but the turbulence is well developed on smaller scales. Study of a neutron star dynamo raises several fundamental issues in the theory of fast dynamos, in particular the possibility of dynamo action in mirror-symmetric turbulence. Other issues include the role played by turbulent diffusion, the relevance of the Bondi-Gold theorem, and the degree of intermittency of the field generated. We argue that in any high-R(m). dynamo, most of the magnetic energy becomes concentrated in thin flux ropes when the field pressure exceeds the turbulent pressure at the smallest scale of turbulence. Most of the magnetic energy of a young pulsar probably resides on scales smaller than the approximately 1 km dimension of an individual convective cell in the newborn neutron star. The surface field strength significantly exceeds that expected in a simple dipole model if even a tiny fraction of the field generated during the convective epoch is retained. The crustal field of a young pulsar should have many discontinuities, at which reconnection is inhibited by the stable stratification of the star. Diffusive processes in the crust eventually allow the field to reconnect, which may result in persistent, detectable shifts in the spin-down rate. The field should provide sufficient free energy to power the largest observed pulsar glitches. We also examine the possibilities for dynamo action during the various (precollapse) stages of convective motion that occur in the evolution of a massive star, and contrast the properties of white dwarf and neutron star progenitors. In general, the Rossby number of the convective motions, and the energy density in these motions, both increase as evolution progresses. Thus, the earliest stages of convection provide the most suitable site for an alpha - OMEGA dynamo, while the latest stages of convection generate the most intense fields. On energetic grounds, main-sequence convective episodes are capable of accounting for a neutron star dipole field stronger than 10(12)-3 x 10(13) G. If the dipole field is generated after collapse, then this field strength also arises naturally as the random superposition of many small dipoles of strength 10(14)-10(15) G and size approximately 1 km. The postcollapse convection should destroy any preexisting correlation between the magnetic and rotation axes. Magnetically induced fluctuations in the brightness of the neutrinosphere will also impart a recoil to the star, which plausibly is of magnitude approximately 100 km s-1. We consider the effect of the strong magnetic fields generated in a nascent neutron star on a supernova explosion, including deposition of energy outside the star via the reaction v --> v + e+ + e-, via neutron scattering off electron pairs trapped in magnetic flux ropes, and also via more conventional mechanisms such as magnetic reconnection and MHD waves. Some of these energy sources may suffice to revive a stalled supernova shock, but only under optimistic assumptions concerning the strength of the dynamo-generated field. We also briefly discuss the possible role of extremely strong magnetic fields in producing gamma-ray bursts.
