In the standard cosmological scenario, galaxies form naturally out of gas and dark matter in the young universe. Then, over the course of their life, galaxies accrete gas from their surroundings and convert it into new stars, thus growing in mass and size. This model is able to explain many of the observed properties of galaxies, from our own Milky Way to the most distant galaxies that are observed when the universe was still in its infancy. However, there is one observation that has puzzled theorists for decades: when galaxies become sufficiently large, they suddenly stop forming stars and cease their growth. In order to reproduce this observation, the standard model needs a key additional ingredient called galaxy quenching, i.e. a physical mechanism that shuts down star formation and transform massive galaxies into quiescent systems. The most popular explanation for quenching is that it is due to the large amount of energy released by the infall of material onto the supermassive black hole that is found at the center of every massive galaxy. However, the details for how this so-called feedback is able to stop the formation of new stars are still unclear, and observational studies have been unable to present direct evidence causally connecting supermassive black hole feedback with galaxy quenching. There are also proposed quenching mechanisms that do not involve black holes, but are based on other processes such as galaxy mergers or gravitational effects. Clearly, identifying the physical mechanism responsible for quenching is crucial for our understanding of how galaxies form and evolve in the context of the standard cosmological scenario.