  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)
