Binary systems comprising black holes and or neutron stars represent the strongest sources of gravitational waves and have recently been revealed by detectors such as LIGO and Virgo. In this series of lectures I will provide a brief introduction to the mathematical and numerical techniques that are needed for the theoretical modelling of these systems. At the same time, I will discuss how neutron-star binaries represent fantastic tools to explore fundamental aspects of gravitational and particle physics. In particular, I will provide a few examples of the ways in which neutron stars can be used to explore fundamental physics, ranging from their spectral properties, the possibility of phase transition to quark matter, the amplification of super-strong magnetic fields, and the copious emission of neutrinos.