One of the key open questions in extragalactic astronomy is what stops star formation in galaxies. While it is clear that the cold gas reservoir, which fuels the formation of new stars, must be affected first, how this happens and what are the dominant physical mechanisms involved is still a matter of debate. At least for satellite galaxies, it is generally accepted that internal processes alone cannot be responsible for fully quenching their star formation, but that environment should play an important, if not dominant, role. In nearby clusters, we see examples of cold gas being removed from the star-forming disks of galaxies moving through the intracluster medium, but whether active stripping is widespread and/or necessary to halt star formation in satellites, or quenching is just a consequence of the inability of these galaxies to replenish their cold gas reservoirs, remains unclear. In this work, we review the current status of environmental studies of cold gas in star-forming satellites in the local Universe from an observational perspective, focusing on the evidence for a physical link between cold gas stripping and quenching of the star formation. We find that stripping of cold gas is ubiquitous in satellite galaxies in both group and cluster environments. While hydrodynamical mechanisms such as ram pressure are important, the emerging picture across the full range of dark matter halos and stellar masses is a complex one, where different physical mechanisms may act simultaneously and cannot always be easily separated. Most importantly, we show that stripping does not always lead to full quenching, as only a fraction of the cold gas reservoir might be affected at the first pericentre passage. We argue that this is a key point to reconcile apparent tensions between statistical and detailed analyses of satellite galaxies...(abridged)
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