Satellite galaxies in groups and clusters aremore likely to have low star formation rates (SFRs) and lie on the 'red sequence' than central ('field') galaxies. Using galaxy group/cluster catalogues from the Sloan Digital Sky Survey Data Release 7, together with a high-resolution, cosmological N-body simulation to track satellite orbits, we examine the star formation histories and quenching time-scales of satellites of Mstar > 5 × 109 M⊙ at z ≈ 0. We first explore satellite infall histories: group preprocessing and ejected orbits are critical aspects of satellite evolution, and properly accounting for these, satellite infall typically occurred at z ~ 0.5, or ~5 Gyr ago. To obtain accurate initial conditions for the SFRs of satellites at their time of first infall, we construct an empirical parametrization for the evolution of central galaxy SFRs and quiescent fractions.With this, we constrain the importance and efficiency of satellite quenching as a function of satellite and host halo mass, finding that satellite quenching is the dominant process for building up all quiescent galaxies at Mstar < 1010M⊙. We then constrain satellite star formation histories, finding a 'delayed-then-rapid' quenching scenario: satellite SFRs evolve unaffected for 2-4 Gyr after infall, after which star formation quenches rapidly, with an e-folding time of <0.8Gyr. These quenching time-scales are shorter for more massive satellites but do not depend on host halo mass: the observed increase in the satellite quiescent fraction with halo mass arises simply because of satellites quenching in a lower mass group prior to infall (group preprocessing), which is responsible for up to half of quenched satellites in massive clusters. Because of the long time delay before quenching starts, satellites experience significant stellar mass growth after infall, nearly identical to central galaxies. This fact provides key physical insight into the subhalo abundance matching method. © 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.