Origin and evolution of neutron star magnetic fields

This paper intends to give a broad overview of the present knowledge about neutron star magnetic fields, their origin and evolution. An up-to-date overview of the rich phenomenology (encompassing ``classical'' and millisecond radio pulsars, X-ray binaries, ``magnetars'', and ``thermal emitters'') suggests that magnetic fields on neutron stars span at least the range $10^{8-15}$ G, corresponding to a range of magnetic fluxes similar to that found in white dwarfs and upper main sequence stars. The limitations of the observational determinations of the field strength and evidence for its evolution are discussed. Speculative ideas about the possible main-sequence origin of the field (``magnetic strip-tease'') are presented. Attention is also given to physical processes potentially leading to magnetic field evolution.