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.