We combine the galaxy formation model GALFORM with the photon-dominated region code UCL_PDR to study the emission from the rotational transitions of 12CO (CO) in galaxies from z = 0 to z = 6 in the Λcold dark matter framework. GALFORM is used to predict the molecular (H 2) and atomic hydrogen (H i) gas contents of galaxies using the pressure-based empirical star formation relation of Blitz & Rosolowsky. From the predicted H 2 mass and the conditions in the interstellar medium, we estimate the CO emission in the rotational transitions 1-0 to 10-9 by applying the UCL_PDR model to each galaxy. We find that deviations from the Milky Way CO-H 2 conversion factor come mainly from variations in metallicity, and in the average gas and star formation rate surface densities. In the local universe, the model predicts a CO(1-0) luminosity function (LF), CO-to-total infrared (IR) luminosity ratios for multiple CO lines and a CO spectral line energy distribution (SLED) which are in good agreement with observations of luminous and ultra-luminous IR galaxies. At high redshifts, the predicted CO SLED of the brightest IR galaxies reproduces the shape and normalization of the observed CO SLED. The model predicts little evolution in the CO-to-IR luminosity ratio for different CO transitions, in good agreement with observations up to z ∈ 5. We use this new hybrid model to explore the potential of using colour-selected samples of high-redshift star-forming galaxies to characterize the evolution of the cold gas mass in galaxies through observations with the Atacama Large Millimetre Array. © 2012 The Authors Monthly Notices of the Royal Astronomical Society.