Since the 1930s, astronomical observations have accumulated evidence that our understanding of the dynamics of galaxies and groups of galaxies is grossly incomplete: assuming the validity of Newton’s law of gravity on astronomical scales, the observed mass (stored in stars and interstellar gas) of stellar systems can account only for roughly 10% of the dynamical (gravitating) mass required to explain the high velocities of stars in those systems. The standard approach to this ‘missing mass problem’ has been the postulate of ‘dark matter’, meaning an additional, electromagnetically dark, matter component that provides the missing mass. However, direct observational evidence for dark matter has not been found to date. More importantly, astronomical observations obtained during the last decade indicate that dark matter cannot explain the kinematics of galaxies. Multiple observations show that the discrepancy between observed and dynamical mass is a function of gravitational acceleration (or field strength) but not of other parameters (size, rotation speed, etc.) of a galaxy; the mass discrepancy appears below a characteristic and universal acceleration aM = (1:1±0:1) · 10 -10 ms -2 (Milgrom’s constant). Consequently, the idea of a modified law of gravity, specifically the ansatz of modified Newtonian dynamics (MOND), is becoming increasingly important in astrophysics. MOND has successfully predicted various important empirical relations of galaxy dynamics, including the famous Tully-Fisher and Faber-Jackson relations. MOND is found to be consistent with stellar dynamics from binary stars to clusters of galaxies, thus covering stellar systems spanning eight orders of magnitude in size and 14 orders of magnitude in mass. These developments have the potential to initiate a paradigm shift from dark matter to a modified law of gravity as the physical mechanism behind the missing mass problem.