  This doctoral thesis investigates the long-term evolution of the strong
magnetic fields within isolated neutron stars (NSs), the most potent magnetic
objects in the universe. Their magnetic influence extends beyond their surface
to encompass the magnetised plasma in their vicinity. The overarching magnetic
configuration significantly impacts the observable characteristics of the
highly magnetised NSs, i.e., magnetars. Conversely, the internal magnetic field
undergoes prolonged evolution spanning thousands to millions of years,
intricately linked to thermal evolution. The diverse observable phenomena
associated with NSs underscore the complex 3D nature of their magnetic
structure, thereby requiring sophisticated numerical simulations. A central
focus of this thesis involves a thorough exploration of state-of-the-art 3D
coupled magneto-thermal evolution models. This marks a pioneering achievement
as we conduct, for the first time, the most realistic 3D simulations to date,
spanning the first million years of a NS's life using the newly developed code
MATINS, which adeptly accounts for both Ohmic dissipation and Hall drift within
the NS's crust. Our simulations incorporate highly accurate
temperature-dependent microphysical calculations and adopt the star's structure
based on a realistic equation of state. To address axial singularities in 3D
simulations, we employ the cubed-sphere coordinates. We also account for
corresponding relativistic factors in the evolution equations and use the
latest envelope model from existing literature, in addition to an initial
magnetic field structure derived from proton-NS dynamo simulations. Within this
framework, we quantitatively simulate the thermal luminosity, timing
properties, and magnetic field evolution, pushing the boundaries of numerical
modeling capabilities and enabling the performance of several astrophysical
studies within this thesis.
