The ASTRI project represents an effort of an international collaboration led by INAF in conjunction with the Universidade de Sao Paulo and the North-West University (South Africa), aiming at assessing the technology of an array of small-size Cherenkov telescopes (SSTs). In this respect, the ASTRI-Horn telescope, an end-to-end prototype, has been installed in Sicily, on the slopes of the Etna volcano at 1735 m a.s.l. The telescope is characterized by a dual-mirror Schwarzschild-Couder design that covers a wide field of view and favors gamma-ray observations in the range from few TeVs up to tens of TeV. Its curved focal-plane camera consists of Silicon-photomultipliers, which are controlled by fast front-end electronics. The complete end-to-end approach includes the implementation of calibration as well as control and acquisition systems, together with the data reduction, archiving and analysis software. The ASTRI collaboration has successfully completed the commissioning and science verification phase of the prototype, including observation of the Crab Nebula. A Mini-Array composed of nine ASTRI telescopes is being developed and operated by INAF in the context of the preparatory effort for the proposed participation in the Cherenkov Telescope Array (CTA). Apart from a few minor changes implemented to improve the telescope design, the mini-array telescopes will be very similar to the ASTRI-Horn telescope. In this contribution an overview of the main features and performance of the ASTRI-Horn telescope is presented together with recent updates and first scientific results. In addition to this, the technological evolution towards the ASTRI mini-array and its scientific expectations are outlined.