  Neutron stars host the strongest magnetic fields that we know of in the
Universe. Their magnetic fields are the main means of generating their
radiation, either magnetospheric or through the crust. Moreover, the evolution
of the magnetic field has been intimately related to explosive events of
magnetars, which host strong magnetic fields, and their persistent thermal
emission. The evolution of the magnetic field in the crusts of neutron stars
has been described within the framework of the Hall effect and Ohmic
dissipation. Yet, this description is limited by the fact that the Maxwell
stresses exerted on the crusts of strongly magnetised neutron stars may lead to
failure and temperature variations. In the former case, a failed crust does not
completely fulfil the necessary conditions for the Hall effect. In the latter,
the variations of temperature are strongly related to the magnetic field
evolution. Finally, sharp gradients of the star's temperature may activate
battery terms and alter the magnetic field structure, especially in weakly
magnetised neutron stars. In this review, we discuss the recent progress made
on these effects. We argue that these phenomena are likely to provide novel
insight into our understanding of neutron stars and their observable
properties.
