The Cherenkov Telescope Array (CTA) is a large collaborative effort aimed at the design and operation of an observatory dedicated to very high-energy gamma-ray astrophysics in the energy range from a few tens of GeV to above 100 TeV, which will yield about an order of magnitude improvement in sensitivity with respect to the current major arrays (H.E.S.S., MAGIC, and VERITAS). Within this framework, the Italian National Institute for Astrophysics is leading the ASTRI project, whose main goals are the design and installation on Mt. Etna (Sicily) of an end-to-end dual-mirror prototype of the CTA small size telescope (SST) and the installation at the CTA Southern site of a dual-mirror SST mini-array composed of nine units with a relative distance of about 300 m. The innovative dual-mirror Schwarzschild-Couder optical solution adopted for the ASTRI Project allows us to substantially reduce the telescope plate-scale and, therefore, to adopt silicon photo-multipliers as light detectors. The ASTRI mini-array is a wider international effort. The mini-array, sensitive in the energy range 1-100 TeV and beyond with an angular resolution of a few arcmin and an energy resolution of about 10-15%, is well suited to study relatively bright sources (a few $times 10^{-12}$erg cm$^{-2}$s$^{-1}$ at 10 TeV) at very high energy. Prominent sources such as extreme blazars, nearby well-known BL Lac objects, Galactic pulsar wind nebulae, supernovae remnants, micro-quasars, and the Galactic Center can be observed in a previously unexplored energy range. The ASTRI mini-array will extend the current IACTs sensitivity well above a few tens of TeV and, at the same time, will allow us to compare our results on a few selected targets with those of current (HAWC) and future high-altitude extensive air-shower detectors.