An asymmetry between left- and right-chiral lepton states can induce
additional contributions to the electromagnetic current in the presence
of magnetic fields or vortical fluid flows. In contrast to the usual Ohmic
current these chiral currents are dissipationless. They can lead to instabilities
such as the chiral magnetic effect in which magnetic fields can undergo an
exponential growth phase, similar to the dynamo effect. We discuss astrophysical
and cosmological contexts in which such effects may play a role, such as
core collapse supernovae and magnetogenesis in the early Universe. Many
open questions, however, remain some of which will also be discussed.