The ALICE experiment was proposed in 1993, to study strongly interacting matter at extreme energy densities via a comprehensive investigation of nuclear collisions at the LHC. Its physics programme initially focused on the determination of the properties of the Quark-Gluon Plasma (QGP), a deconfined state of quarks and gluons and was extended along the years, covering a diverse ensemble of observables related to Quantum Chromodynamics (QCD), the theory of strong interactions. The experiment has studied Pb-Pb, Xe-Xe, p-Pb and pp collisions in the multi-TeV energy range, during the Run 1 and Run 2 data taking periods at the LHC (2009-2018). The aim of this review article is to gather and summarise a selection of ALICE physics results and to discuss their implications on the current understanding of the macroscopic and microscopic properties of strongly interacting matter at the highest temperature reached in the laboratory. It will be shown that it is possible to have a quantitative description of the properties of the QGP produced in Pb--Pb collisions. We also show that various features, commonly ascribed to QGP formation, are detected for a wide range of interacting system sizes. Precision measurements of QCD-related observables not directly connected to the study of the QGP will also be discussed. Prospects for future measurements with the ALICE detector and its foreseen upgrades will also be briefly described.