The principle of relativity requires that no interaction can propagate instantly. Gravitational waves (GW) must exist, propagating with the same speed as light. The specific characters of GW are predicted by Einstein’s general relativity (GR). After decades of efforts to develop detectors, on 11 February 2016, the LIGO and Virgo Collaboration published the discovery of a GW.

The elements of GR relevant for GW production, propagation and detection. How the GR field, which is the dimensionless metric tensor, differs from the other fundamental fields, which have physical dimensions. The instruments and the discovery. After the first observation, dozens of gravitational signals have been detected, the vast majority from merging black holes and one, on 17 August 2017, from the merger of neutron stars. In this case, electromagnetic signals are expected, and have been detected, providing unique information to astrophysics and to fundamental physics as well. The measurement of the speed of the GW and the establishment of a bound on the mass of the graviton.

The detection of gravitational waves from the merger of black holes and the merger of two neutron stars are discussed. The linearized theory of general relativity is introduced. The concept of the gauge invariance is put forward and the transverse-traceless gauge for gravitational waves is presented. Einstein‘s famous quadrupole formula for gravitational waves is developed. The principle of detecting gravitational waves is outlined. As applications, the emission of gravitational waves from a nonvanishing ellipticity of rotating neutron stars is derived. The chirp mass is introduced and the emission of gravitational waves from compact binary systems is obtained. The formula for the tidal deformability and the Love number is put forward and discussed with regard to the recent measurement of a neutron star merger by the LIGO–Virgo scientific collaboration.