  Neutron stars are natural physical laboratories allowing us to study a
plethora of phenomena in extreme conditions. In particular, these compact
objects can have very strong magnetic fields with non-trivial origin and
evolution. In many respects its magnetic field determines the appearance of a
neutron star. Thus, understanding the field properties is important for
interpretation of observational data. Complementing this, observations of
diverse kinds of neutron stars enable us to probe parameters of
electro-dynamical processes at scales unavailable in terrestrial laboratories.
In this review we first briefly describe theoretical models of formation and
evolution of magnetic field of neutron stars, paying special attention to field
decay processes. Then we present important observational results related to
field properties of different types of compact objects: magnetars, cooling
neutron stars, radio pulsars, sources in binary systems. After that, we discuss
which observations can shed light on obscure characteristics of neutron star
magnetic fields and their behaviour. We end the review with a subjective list
of open problems.
