Understanding the process responsible for transforming star forming galaxies into passive and quiescent systems is currently one of the hottest topics in astronomy. I will discuss recent observational results probing different mechanisms at work in different galaxies and at different epochs. Multi-wavelength observations have provided clear evidence that powerful starburst-driven and AGN-driven outflows, both locally and at high-z, can remove large amount of gas out of their host galaxies and, therefore, have been suggested to be the main mechanism responsible for quenching star formation in galaxies. However, more detailed observations, along with detailed simulations, have shown that such massive outflows may not be able to completely quench star formation in galaxies. I will show that the analysis of the stellar metallicities in large samples of local galaxies reveals that “starvation" (or sometimes called “strangulation”, i.e. the lack of gas inflows), is actually the primary mechanism responsible for quenching star formation in most galaxies. I will discuss the possible mechanisms responsible for the starvation of galaxies. I will finally discuss that neither of these mechanisms is anyhow enough to keep the galaxy quiescent, and a continuous "maintenance mode" must be present in galaxies to prevent gas returned into the galaxy, by simple stellar evolution, from forming new stars.