Aims: Our scientific goal is to provide revised membership lists of the Alpha Per, Pleiades, and Praesepe clusters exploiting the second data release of Gaia and produce five-dimensional maps (ra, dec, parallax, pmRA, pmDEC) 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 the Alpha 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 Alpha Per, Pleiades, and Praesepe clusters from the most massive stars all the way down into 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. ABRIDGED
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