We use a coupled model of the formation and evolution of galaxies and black holes (BHs) to study the evolution of active galactic nuclei (AGNs) in a cold dark matter universe. The model is embedded in the galaxy formation code GALFORM and predicts the masses, spins and mass accretion histories of BHs in tandem with the formation of their host galaxies. BHs grow by accretion during starbursts, triggered by discs becoming dynamically unstable or by galaxy mergers, and accretion from quasi-hydrostatic hot gas haloes. Using an empirical law for AGN obscuration, our model matches the observed luminosity functions (LFs) of AGNs over a wide range of redshifts. Due to the suppression of cooling in massive haloes by AGN feedback, at low redshift ( z (cid:2) 2), the brightest quasars ( L bol (cid:3) 10 46 erg s − 1 ) are predicted preferentially to inhabit haloes with masses (cid:2) 10 12 –10 13 M (cid:3) . The model predicts a hierarchical buildup of BH mass, with the typical mass of actively growing BHs increasing with decreasing redshift. Nevertheless, the model displays clear ‘downsizing’ as reflected in the differential evolution of the space density of faint and bright AGNs. This arises naturally from the interplay between the starburst and hot gas halo accretion modes. The faint end of the LF is dominated by massive BHs accreting at low rates via a thick disc, primarily during the hot-halo mode. The bright end is populated by BHs accreting close to or above the Eddington limit during the starburst mode. Obscuration plays a central role in determining the observed abundance of AGNs and, hence, in their implied cosmic evolution.