Aims: Our scientific goal is to provide revised membership lists of the α Per, Pleiades, and Praesepe clusters exploiting the second data release of Gaia and produce five-dimensional maps (α, δ, π, μα cos δ, μδ) of these clusters. Methods: We implemented the kinematic method combined with the statistical treatment of parallaxes and proper motions to identify astrometric member candidates of three of the most nearby and best studied open clusters in the sky. Results: We cross-correlated the Gaia catalogue with large-scale public surveys to complement the astrometry of Gaia with multi-band photometry from the optical to the mid-infrared. We identified 517, 1248, and 721 bona fide astrometric member candidates inside the tidal radius of α Per, the Pleiades, and Praesepe, respectively. We cross-matched our final samples with catalogues from previous surveys to address the level of completeness. We update the main physical properties of the clusters, including mean distance and velocity, as well as core, half-mass, and tidal radii. We infer updated ages from the white dwarf members of the Pleiades and Praesepe. We derive the luminosity and mass functions of the three clusters and compare them to the field mass function. We compute the positions in space of all member candidates in the three regions to investigate their distribution in space. Conclusions: We provide updated distances and kinematics for the three clusters. We identify a list of members in the α Per, Pleiades, and Praesepe clusters from the most massive stars all the way down to the hydrogen-burning limit with a higher confidence and better astrometry than previous studies. We produce complete 5D maps of stellar and substellar bona fide members in these three regions. The photometric sequences derived in several colour-magnitude diagrams represent benchmark cluster sequences at ages from 90 to 600 Myr. We note the presence of a stream around the Pleiades cluster extending up to 40 pc from the cluster centre. Full Tables A.1, B.1, and C.1 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/628/A66