Classical general relativity is usually referred as one of the most beautiful physical theories and one reason for this claim is that it admits an elegant geometrically inspired initial value formulation, with deep connections to physical reality. After tracing the main ideas that led to the current view of understanding, I will present the main formulations that lead to astrophysically important solutions both in vacuum as well as in the presence of matter. The numerical implementation of these methods will be presented in the context of the Compact Object CALculator (COCAL) code whose purpose is to solve for any general-relativistic system in equilibrium or quasi-equilibrium. This includes rotating stars, binary neutron stars, binary black holes, black hole-toroids and many others. Future prospects will be discussed.