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The quenching of star formation activity represents a critical phase for a non-negligible fraction of the observed galaxy population at all cosmic epochs, marking a transition from an epoch of intense mass growth to an extended period of passive evolution. Over the past years, we have collected a detailed characterization of how the fraction of quenched galaxies evolves as a function of cosmic time (it grows at later cosmic epochs), and correlates with the physical properties of galaxies (more massive galaxies, that also tend to be bulge-dominated, are more likely to be quenched) and with their environment (denser environments host larger fractions of quiescent galaxies). Different physical processes can lead to a suppression of star formation. These include internal processes (processes that do not depend on the environment in which galaxies live, e.g., stellar and AGN feedback mechanisms) and processes that instead depend on the environment (e.g., mergers, gas stripping processes). In this chapter, we summarize the observational and theoretical status of the field, highlighting the most recent results and the questions that are yet to be answered. We also summarize our view on the expected developments in the next few years.