Neutron stars are an excellent laboratory in which to study fundamental physics in regimes that cannot be probed in terrestrial laboratories. Recent observations of gravitational waves from neutron star binaries have opened a new window for multi-messenger studies of these extreme objects, and future observations, possibly of still undetected signals such as continuous gravitational waves waves will allow to probe the interior with greater precision. To enable such constraints, theoretical studies are however necessary, and I will focus in particular on our studies of superfluidity and superconductivity in the interior of the star. I will describe the role that quantum turbulence may have in vortex motion in neutron stars (that can be studied by timing radio pulsars), but also its impact on gravitational wave detection. I will also discuss the impact of superconductivity on the magnetic field, and how MHD turbulence may play a role in setting its large scale topology.