We examine the evolution of intrinsic u-r colours of galaxies in the EAGLE cosmological hydrodynamical simulations, which has been shown to reproduce the observed redshift z = 0.1 colour-magnitude distribution well, with a focus on z < 2. The median u-r of star-forming (`blue cloud') galaxies reddens by 1 mag from z = 2 to 0 at fixed stellar mass, as their specific star formation rates decrease with time. A red sequence starts to build-up around z = 1, due to the quenching of low-mass satellite galaxies at the faint end, and due to the quenching of more massive central galaxies by their active galactic nuclei (AGN) at the bright end. This leaves a dearth of intermediate-mass red sequence galaxies at z = 1, which is mostly filled in by z = 0. We quantify the time-scales of colour transition finding that most galaxies spend less than 2 Gyr in the `green valley'. We find the time-scale of transition to be independent of quenching mechanism, i.e. whether a galaxy is a satellite or hosting an AGN. On examining the trajectories of galaxies in a colour-stellar mass diagram, we identify three characteristic tracks that galaxies follow (quiescently star-forming, quenching and rejuvenating galaxies) and quantify the fraction of galaxies that follow each track.