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We review methods that allow one to detect and characterise quantumcorrelations in many-body systems, with a special focus on approaches which arescalable. Namely, those applicable to systems with many degrees of freedom,without requiring a number of measurements or computational resources toanalyze the data that scale exponentially with the system size. We begin withintroducing the concepts of quantum entanglement, Einstein-Podolsky-Rosensteering, and Bell nonlocality in the bipartite scenario, to then present theirmultipartite generalisation. We review recent progress on characterizing thesequantum correlations from partial information on the system state, such asthrough data-driven methods or witnesses based on low-order moments ofcollective observables. We then review state-of-the-art experiments thatdemonstrated the preparation, manipulation and detection of highly-entangledmany-body systems. For each platform (e.g. atoms, ions, photons,superconducting circuits) we illustrate the available toolbox for statepreparation and measurement, emphasizing the challenges that each system poses.To conclude, we present a list of timely open problems in the field.

Probing quantum correlations in many-body systems: a review of scalable methods