  To be observed and analyzed by the network of current gravitational wave
detectors (LIGO, Virgo, KAGRA), and in anticipation of future third generation
ground based (Einstein Telescope, Cosmic Explorer) and space borne (LISA)
detectors, inspiralling compact binaries -- binary star systems composed of
neutron stars and/or black holes in their late stage of evolution prior the
final coalescence -- require high-accuracy predictions from general relativity.
The orbital dynamics and emitted gravitational waves of these very relativistic
systems can be accurately modelled using state-of-the-art post-Newtonian
theory. In this article we review the Multipolar-Post-Minkowskian approximation
scheme, merged to the standard Post-Newtonian expansion into a single formalism
valid for general isolated matter system. This cocktail of approximation
methods (called MPM-PN) has been successfully applied to compact binary
systems, producing equations of motion up to the fourth-post-Newtonian (4PN)
level, and gravitational waveform and flux to 4.5PN order beyond the Einstein
quadrupole formula. We describe the dimensional regularization at work in such
high post-Newtonian calculations, for curing both ultra-violet and infra-red
divergences. Several landmark results are detailed: the definition of multipole
moments, the gravitational radiation reaction, the conservative dynamics of
circular orbits, the first law of compact binary mechanics, and the non-linear
effects in the gravitational wave propagation (tails, iterated tails and
non-linear memory). We also discuss the case of compact binaries moving on
eccentric orbits, and the effects of spins (both spin-orbit and spin-spin) on
the equations of motion and gravitational wave energy flux and waveform.
