Sco OB2 is the nearest OB association, extending over approximately 2000 sq.deg. on the sky. Only its brightest members are already known (from Hipparcos) across its entire size, while studies of its lower-mass population refer only to small portions of its extent. In this work we exploit the capabilities of Gaia DR2 measurements to search for Sco OB2 members across its entire size and down to the lowest stellar masses. We use both Gaia astrometric and photometric data to select association members, using minimal assumptions derived mostly from the Hipparcos studies. Gaia resolves small details in both the kinematics of individual Sco OB2 subgroups and their distances from the Sun. We develop methods to explore the 3D kinematics of stellar populations covering large sky areas. We find ~11000 pre-main sequence (PMS) Sco OB2 members (with <3% contamination), plus ~3600 MS candidate members with a larger (10-30%) field-star contamination. A higher-confidence subsample of ~9200 PMS (and ~1340 MS) members is also selected (<1% contamination for the PMS), affected however by larger (~15%) incompleteness. We classify separately stars in compact and diffuse populations. Most members belong to a few kinematically distinct diffuse populations, whose ensemble outlines the association shape. Upper Sco is the densest part of Sco OB2, with a complex spatial and kinematical structure, and no global pattern of motion. Other dense subclusters are found in Upper Centaurus-Lupus and in Lower Centaurus-Crux. Most clustered stars appear to be younger than the diffuse PMS population, suggesting star formation in small groups which rapidly disperse and dilute, while keeping memory of their original kinematics. We also find that the open cluster IC 2602 has a similar dynamics to Sco OB2, and its PMS members are evaporating and forming a ~10 deg halo around its double-peaked core.
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