The article reviews the current status of a theoretical approach to the emission of gravitational waves by isolated systems in general relativity. It discusses post-Newtonian sources and their radiative multipole moments, and applies this to compact binary systems. The post-Newtonian approximation is valid for weak gravitational fields and slow internal motions, and is used to model neutron stars and black holes. The quadrupole formalism, based on Newtonian gravity, is a key tool for calculating gravitational wave emission. However, it is limited to the near zone of the source and does not account for tail effects. The post-Minkowskian expansion is more fundamental and uniformly valid for weakly self-gravitating sources. The article also discusses the multipole expansion, which is useful for describing the gravitational field of a source in the far zone. The post-Newtonian approximation is used to model compact binaries, and the equations of motion are derived up to the third post-Newtonian order. The gravitational wave energy flux is calculated up to the 3.5PN order, taking into account relativistic corrections and tail effects. The orbital phase of the binary is determined from an energy balance argument. The article also discusses the post-Newtonian equations of motion and radiation, showing that the 3PN approximation is necessary for accurate modeling of gravitational wave signals from compact binaries. The results are validated by comparing them with other methods, such as linear black-hole perturbations. The article concludes that the 3PN approximation is essential for understanding the non-linear structure of general relativity and for the detection of gravitational waves from compact binaries.The article reviews the current status of a theoretical approach to the emission of gravitational waves by isolated systems in general relativity. It discusses post-Newtonian sources and their radiative multipole moments, and applies this to compact binary systems. The post-Newtonian approximation is valid for weak gravitational fields and slow internal motions, and is used to model neutron stars and black holes. The quadrupole formalism, based on Newtonian gravity, is a key tool for calculating gravitational wave emission. However, it is limited to the near zone of the source and does not account for tail effects. The post-Minkowskian expansion is more fundamental and uniformly valid for weakly self-gravitating sources. The article also discusses the multipole expansion, which is useful for describing the gravitational field of a source in the far zone. The post-Newtonian approximation is used to model compact binaries, and the equations of motion are derived up to the third post-Newtonian order. The gravitational wave energy flux is calculated up to the 3.5PN order, taking into account relativistic corrections and tail effects. The orbital phase of the binary is determined from an energy balance argument. The article also discusses the post-Newtonian equations of motion and radiation, showing that the 3PN approximation is necessary for accurate modeling of gravitational wave signals from compact binaries. The results are validated by comparing them with other methods, such as linear black-hole perturbations. The article concludes that the 3PN approximation is essential for understanding the non-linear structure of general relativity and for the detection of gravitational waves from compact binaries.