Observing the neutral hydrogen in galaxy clusters provides crucial insights in the physical processes that influence the evolution of gas-rich galaxies as they migrate from the lower-density filaments through the cluster outskirts into to the higher-density central regions. The morphology-density relation, the Butcher-Oemler effect, and the observed HI deficiencies in the central regions of galaxy clusters suggest that infalling galaxies experience a strong transformation of their morphologies, star formation rates, and gas content. The physical mechanisms that trigger and govern these transformations may depend strongly on environment. This contribution aims to illustrate that the morphological and kinematic characteristics of the cold gas provide a sensitive tool to determine which mechanisms dominate in which environments.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html