Typically, in studies of the quasi-stationary evolution of the magnetic field in neutron stars (NSs), the main attention is paid to investigating the magnetic field in the star’s crust (a relatively thin, 1 km, near-surface layer whose microphysics is relatively well understood). Most such studies implicitly assume that the magnetic field in the inner regions of the NS (in its core) either does not exist or its evolutionary timescales are so large that it can be considered ``frozen’’. In the first part of the talk, I will show that the role of the core is often unjustifiably ignored and that there exists an effect that can significantly accelerate the field evolution in the core. In the second part of the talk, I will discuss our new results related to the study of instabilities that arise in magnetized nonrotating neutron stars. The main attention will be given to the discussion of an instability of the toroidal field, which is important for applications. Using numerical and analytical models, we have studied in detail the mechanism of the development of this instability. It is shown that it is global, and thus differs from the local instability predicted by Tayler in 1973.