  We present new bounds on the cosmic abundance of magnetic monopoles based on
the survival of primordial magnetic fields during the reheating and
radiation-dominated epochs. The new bounds can be stronger than the
conventional Parker bound from galactic magnetic fields, as well as bounds from
direct searches. We also apply our bounds to monopoles produced by the
primordial magnetic fields themselves through the Schwinger effect, and derive
additional conditions for the survival of the primordial fields.
