Numerical Relativity is a multidisciplinary field including relativity, magneto-hydrodynamics, astrophysics and computational methods, among others, with the aim of solving numerically highly-dynamical, strong-gravity scenarios where no other approximations are available. Here we describe some of the foundations of the field, starting from the covariant Einstein equations and how to write them as a well-posed system of evolution equations, discussing the different formalisms, coordinate conditions and numerical methods commonly employed nowadays for the modeling of gravitational wave sources.

Accepted by Frontiers Astronomy and Space Sciences, invited review for the Research Topic "Gravitational Waves: A New Window to the Universe"