The r-process of nucleosynthesis is the process which is responsible for the synthesis of approximately half of the nuclear species in nature that are more massive than iron. This process of heavy-element synthesis involves the progressive buildup of heavier isotopes via neutron captures proceeding on neutron-rich isotopes, interspersed by beta decays. Its abundance features clearly reflect nuclear properties - the maxima are related to the magic neutron numbers N = 50, 82 and 126. It probes our knowledge of the properties of nuclei far from the region of beta stability, even through the position of the neutron drip line. The r-process also forms the important long-lived nuclear chronometers Th-232, U-238 and U-235 that are utilized for dating the Galaxy. While the astrophysical site for r-process nucleosynthesis is not yet identified, its association with massive stars undergoing type II supernova events is strongly suggested. This can be deduced from the observation that r-process nuclei are already present in the oldest and most metal-deficient stars, which are the tracers of the chemical evolution of the Galaxy. The massive stars that become type II supernova evolve fastest and contribute their ejecta to the interstellar medium al the earliest beginnings of the chemical evolution of the Galaxy. When utilizing the knowledge of the r-process production ratios of the long-lived chronometer nuclei, their observed ratios in primitive meteorites and our (limited) knowledge of star formation throughout galactic evolution, one can put limits on the duration of galactic nucleosynthesis, the age of the Galaxy and the universe. The latter has a large uncertainty but is comparable with ages estimated from globular clusters and cosmology. This paper will provide a review of the basic physics underlying the r-process, the operation of the mechanisms for r-process nucleosynthesis, the possible astrophysical sites, their time evolution in galactic evolution models. and the inferred ages of the Galaxy.