We use SDSS+GALEX+Galaxy Zoo data to study the quenching of star formation in lowredshift galaxies. We show that the green valley between the blue cloud of star-forming galaxies and the red sequence of quiescent galaxies in the colour-mass diagram is not a single transitional state through which most blue galaxies evolve into red galaxies. Rather, an analysis that takes morphology into account makes clear that only a small population of blue earlytype galaxies move rapidly across the green valley after the morphologies are transformed from disk to spheroid and star formation is quenched rapidly. In contrast, the majority of blue star-forming galaxies have significant disks, and they retain their late-type morphologies as their star formation rates decline very slowly. We summarize a range of observations that lead to these conclusions, including UV-optical colours and halo masses, which both show a striking dependence on morphological type. We interpret these results in terms of the evolution of cosmic gas supply and gas reservoirs. We conclude that late-type galaxies are consistent with a scenario where the cosmic supply of gas is shut off, perhaps at a critical halo mass, followed by a slow exhaustion of the remaining gas over several Gyr, driven by secular and/or environmental processes. In contrast, early-type galaxies require a scenario where the gas supply and gas reservoir are destroyed virtually instantaneously, with rapid quenching accompanied by a morphological transformation from disk to spheroid. This gas reservoir destruction could be the consequence of a major merger, which in most cases transforms galaxies from disk to elliptical morphology, and mergers could play a role in inducing black hole accretion and possibly AGN feedback.