Magnetic, thermal and rotational evolution of isolated neutron stars

Abstract The strong magnetic fields of neutron stars are closely linked to their observed thermal, spectral, and timing properties, such as the distribution of spin periods and their derivatives. To understand the evolution of astrophysical observables over time, it is essential to develop robust theoretical frameworks and numerical models that solve the coupled thermal and magnetic field evolution equations, incorporating detailed microphysics such as thermal and electrical conductivities and neutrino emission rates. These efforts are key to uncovering how the strength and geometry of magnetic fields change with age, ultimately shedding light on the diverse phenomenology of neutron stars. In this review, we outline the fundamental theory underlying magneto-thermal evolution models, with an emphasis on numerical methods and a comprehensive set of benchmark tests intended to guide current and future code development. We revisit established results from axisymmetric simulations, highlight recent progress in fully three-dimensional models, and offer a perspective on the anticipated developments in this rapidly evolving field.