The General Theory of Relativity was established by Albert Einstein in 1915, at that time without any empirical basis.
In the following years several experiments were performed to confirm the different predictions of GR. In general, the experiments can be divided into classical and modern ones. One of the classical experiments is about the perihelion rotation of the planet Mercury. Already at the end of the nineteenth century astronomers noticed a discrepancy of 43″ arc seconds in one hundred years between the measurements of the speed of the perihelion rotation and the predictions of the Newtonian theory used at that time. In 1916, only a few years later, Einstein was able to calculate the discrepancy exactly and thus explain it with his General Theory of Relativity, which had been established shortly before. Again three years later the theory had its public breakthrough when astronomer Arthur Eddington in 1919 could show in his experiments that the light of a star is deflected by the gravitational field of the sun and thus space is curved according to Einstein’s theory . Applying Newton’s theory, a curvature of space could be calculated as well, but it would yield a different result than GR. Eddington’s experiment was performed during a solar eclipse, because only then starlight can be observed simultaneously with the eclipsed sun. The setup is shown in the following figure.
Light from a star passing close to the Sun can be observed on Earth when deflected by the Sun’s own gravitational field. Actually, the star itself is not visible as its true position (from the observer’s point of view) is behind the Sun. However, the true position of the star can be calculated and is found to be 1.75 arcseconds using GR. In Eddington’s experiment this result of GR could be confirmed with an uncertainty of 20%. Since the first measurement in 1919, the experiment has been repeated and confirmed many times with a higher accuracy   .
Another result of general relativity was the gravitational redshift, i.e. the change of the light frequency into the red (smaller wavelengths) when light moves from a larger to a smaller gravitational potential. This frequency change can also be calculated with GR. In an experiment of the physicists Pound and Rebka in 1960 the described frequency change could be demonstrated . The setup was in a tower, with a light source with gamma radiation at the bottom (the earth potential) and radiating about 22.6 m vertically upward to the top of the tower (tower potential). Simplified the change of the frequency can be described by
With g being the gravitational acceleration, h the height of the tower and c the speed of light. The uncertainty measured in the experiment corresponded to the theoretical prediction
of about 1 %. By means of the so-called Mößbauer effect it was possible to measure such a small deviation of the frequencies. A red shift is also caused by the expansion of space  .
Shapiro performed another experiment in 1968 and 1971: a radar transit time measurement. Venus, sun and earth lie on a straight line with the distances r1 from the earth to the sun and the distance r2 from Venus to the sun, whereby the Venus, seen from the earth, lies behind the sun   .
A radar signal is now sent from Earth to Venus, reflected there and then registered again on Earth. According to a prediction of GR also the transit time of light is changed when it comes near a large mass. Using Newtonian physics, the transit time can be calculated as follows
Basing on Einstein’s equations following deviation (simplified expression) can be calculated:
Where G is Newton’s gravitational constant, M is the mass of the sun, R is the radius of the sun, and c is the speed of light. In Shapiro’s experiments, the calculated deviation was very well demonstrated experimentally .
In 1971, Hafele and Keating performed an experiment comparing the time displayed on a clock on Earth to that on a clock positioned in an airplane . For an equatorial trajectory, according to the general theory of relativity, the following results are obtained
With τf and τe being the proper times for the flight orbit in the airplane or on the earth, g the gravitational acceleration, h the height of the airplane, real earth radius, e the rotational speed of the earth and c the speed of light. The first term is derived from GR to include gravity, the second term is deduced from special relativity. Theoretically, this results in
which could be measured with the atomic clocks used.
A further, well-known experiment is the Gravity Probe B experiment that was able to prove the Geodesic Effect and the Lense Thirring Effect.
In addition, the equations of GR contain so-called gravitational waves. Like electromagnetic waves these waves propagate at the speed of light, being generated whenever masses are accelerated, e.g. moving planets. The waves‘ intensity is extremely low and additional to that they are affected by many external disturbances. Therefore detection of gravitational waves is a very difficult task. In the 1960s, the physicist Joseph Weber tried to measure the expansion of an aluminum cylinder caused by gravitational waves, but without success  . Sonne, Bernd: Allgemeine Relativitätstheorie für jedermann: Grundlagen, Experimente und Anwendungen verständlich formuliert; Springer Spektrum; 2. Auflage; 2018  Welt der Physik: Albert Einstein und die Relativitätstheorie; (Link: https://www.weltderphysik.de/thema/albert-einstein-und-die-relativitaetstheorie/), accessed 19 November 2020  Fließbach, Thorsten: Allgemeine Relativitätstheorie; Springer Spektrum; 7. Auflage; 2016  Göbel, Holger: Gravitation und Relativität: eine Einführung in die Allgemeine Relativitätstheorie; De Gruyter Studium; 2014  Einstein, Albert: Die Grundlage der allgemeinen Relativitätstheorie; 1923  Eddington, Sir Arthur: Space, Time and Gravitation – An Outline of the General Relativity Theory; Cambridge University Press; 1920  Pound, R. V.; Rebka, G. A. Jr.: Gravitational Red-Shift in Nuclear Resonance; The Physical Review Journals Celebrate The International Year of Light; Phys. Rev. Lett. 3, 439; 01.11.1959  Weber, J.: Gravitational Radiation; Phys. Rev. Lett. 18, 498; 27.03.1967  Ash, M. E.; Campbell, D. B.; Dyce, R. B.; Ingalls, R. P.; Jurgens, R.; Pettengill, G. H.; Shapiro, I. I.: The Case for the Radar Radius of Venus; Science Vol 160, Issue 3831, 31.05.1968  Shapiro, Irwin I.: Radar Observations of the Planets; Scientific American Vol. 219, No. 1; 07/1968  Shapiro, Irwin I.; Ash, Michael E.; Ingalls, Richard P.; Smith, William B.; Campbell, Donald B.; Dyce, Raymond F. Jurgens; Pettengill, Gordon H.: Fourth Test of General Relativity: New Radar Result; Phys. Rev. Lett. 26; 1132; 03.05.1971  Hafele, J. C.; Keating, Richard E.: Around-the-World Atomic Clocks: Predicted Relativistic Time Gains; Science Vol 177, Issue 4044; 14.07.1972  Everitt, C. W. F. et al.: Gravity Probe B: Final Results of a Space Experiment to Test General Relativity; Phys. Rev. Lett. 106, 221101; 31.05.2011
Further reading: Eddington, Sir Arthur: Report on the relativity theory of gravitation; Minkowski Institute Press; 2014  Horst, Wegener: Der Mössbauer-Effekt und seine Anwendungen in Physik und Chemie; Bibliographisches Institut, Mannheim; 1965  Shapiro, Irwin I.; Pettengill, Gordon H.; Ash, Michael E.; Stone, Melvin L.; Smith, William B.; Ingalls, Richard P.; Brockelman, Richard A.: Fourth Test of General Relativity: Preliminary Results; Phys. Rev. Lett. 20, 1265; 27.05.1968, Erratum Phys. Rev. Lett. 21, 266; 1968  Shapiro, Irwin I.: Planetary radar astronomy; IEEE Spectrum Volume 5, Issue 3; 05/1968  Rodrigues Jr., W. A.; de Oliveira, E. C.: A comment on the twin paradox and the Hafele-Keating experiment; Physics Letters A; Volume 140, Issue 9; 16.10.1989  Schlegel, Richard: Comments on the Hafele-Keating Experiment; American Journal of Physics 42, 183; 1974  Everitt, C. W. F.; Mulfelder, B.; DeBra, D. B.; Parkinson, B. W.; Turneaure, J. P.; Silbergleit, A. S.; Acworth, E. B.; Adams, M.; Adler, R.; Bencze, W. J.: The Gravity Probe B test of general relativity; Classical and Quantum Gravity, Volume 32, Number 22; 17.11.2015  Buchman, Saps; Everitt, C. W. F.; Perkinson, B.; et. Al.: The Gravity Probe B Relativity Mission; Advances in Space Research; Volume 25, Issue 6, 2000  Conklin, John W.; Gravity Probe B Collaboration: The Gravity Probe B experiment and early results; Journal of Physics: Conference Series, Volume 140, Issue 1; 2008