While the flat space two-body problem is integrable, the generally-relativistic one is not starting at the next-to-leading order.
In the appropriate classical limit, scattering amplitude-based techniques can yield the classical interaction of massive bodies to all orders in their velocities and to fixed order in the expansion in Newton's constant, that is a fixed order in the post-Minkowskian (relativistic weak-field) expansion.
In this talk we review an amplitudes-based framework for such calculations and the derivation of the third order in the post-Minkowskian  expansion for the conservative Hamiltonian of a compact spinless binary system. We also describe the scattering angle at this order as well as a first comparison with numerical GR.