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Although celestial sources emitting in the few tens of GeV up to a few TeV are being investigated by imaging atmospheric Čerenkov telescope arrays such as H.E.S.S., MAGIC, and VERITAS, at higher energies, up to PeV, more suitable instrumentation is required to detect ultra-high-energy photons, such as extensive air shower arrays, as HAWC, LHAASO, Tibet AS- γ. The Italian National Institute for Astrophysics has recently become the leader of an international project, the ASTRI Mini-Array, with the aim of installing and operating an array of nine dual-mirror Čerenkov telescopes at the Observatorio del Teide in Spain starting in 2025. The ASTRI Mini-Array is expected to span a wide range of energies (1–200 TeV), with a large field of view (about 10 degrees) and an angular and energy resolution of ∼3 arcmin and ∼10 %, respectively. The first four years of operations will be dedicated to the exploitation of Core Science, with a small and selected number of pointings with the goal of addressing some of the fundamental questions on the origin of cosmic rays, cosmology, and fundamental physics, the time-domain astrophysics and non γ -ray studies (e.g., stellar intensity interferometry and direct measurements of cosmic rays). Subsequently, four more years will be dedicated to Observatory Science, open to the scientific community through the submission of observational proposals selected on a competitive basis. In this paper, I will review the Core Science topics and provide examples of possible Observatory Science cases, taking into account the synergies with current and upcoming observational facilities. [ABSTRACT FROM AUTHOR]