The portion of the electromagnetic spectrum above a few Teraelectronvolt (TeV) is currently being investigated by means of both ground-based imaging atmospheric Cherenkov telescopes and water Cherenkov detector arrays. In a few years, an array composed of at least nine ASTRI dual-mirror, Schwarzschild-Couder telescopes will be deployed and start scientific observations at the Observatorio del Teide (Tenerife, Spain). The ASTRI Mini-Array will devote the first three to four observing years to specific science topics, with the aim of providing robust answers to a few selected open questions in the very high-energy (VHE, E > 0.1 TeV) domain. We identified the following Core Science topics, a.k.a. the "Science Pillars", to be investigated: the origin of cosmic rays, the extra-galactic background light and the study of fundamental physics, the novel field in the VHE domain of gamma-ray bursts and other multi-messenger transients, and finally the usage of the ASTRI Mini-Array to investigate ultra high-energy cosmic rays and to address stellar intensity interferometry studies. We review the results obtained by means of dedicated scientific simulations, proving the potential of the ASTRI Mini-Array in pursuing breakthrough discoveries and discuss the synergies with current and future VHE facilities.