I will discuss some general properties of the magnetospheres of pulsars and black holes. Their rich dynamics are responsible for many interesting phenomena, such as the powerful polar jets and fast radio bursts. The magnetospheres are made mainly of a force-free magnetized plasma, where the local electric field is perpendicular to the magnetic field. However, a consistent consideration of the dynamics points to the existence of transient gap regions in the magnetospheres where the projection of the electric field on the magnetic field must be nonzero. Under such conditions, the chiral anomaly can be activated and produce a nonzero chiral charge density in the plasma. In the case of supermassive black holes, the chiral charge is too small to have any observable effects. In contrast, the chiral asymmetry produced in the magnetospheres of magnetars can be substantial. It can trigger chiral plasma instability and possibly lead to observable phenomena in magnetars.