  Cosmic strings are stable topological defects that may have been created at a
phase transition in the early universe. It is a growing belief that, for a wide
range of theoretical models, such strings may be superconducting and carry
substantial currents which have important astrophysical and cosmological
effects. This paper explores the possibility of generation of a primordial
magnetic field by a network of charged-current carrying cosmic strings. The
field is created by vorticity, generated in the primordial plasma due to the
strings' motion and gravitational pull. In the case of superconducting strings
formed at the breaking of grand unification, it is found that strong magnetic
fields of high coherence can be generated in that way. Such fields could
account for the observed galactic and intergalactic magnetic fields since they
suffice to seed magnetic dynamos on galactic scales.
