We use SDSS+GALEX+Galaxy Zoo data to study the quenching of star formation in low-redshift 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 early-type galaxies move rapidly across the green valley after the morphologies are transformed from disc to spheroid and star formation is quenched rapidly. In contrast, the majority of blue star-forming galaxies have significant discs, 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 disc to spheroid. This gas reservoir destruction could be the consequence of a major merger, which in most cases transforms galaxies from disc to elliptical morphology, and mergers could play a role in inducing black hole accretion and possibly active galactic nuclei feedback.