The presence of non-local and long-range interactions in quantum systems induces several peculiar features in their equilibrium and out-of-equilibrium behavior. In current experimental platforms control parameters such as interaction range, temperature, density and dimension can be changed. The existence of universal scaling regimes, where diverse physical systems and observables display quantitative agreement, generates a common framework, where the efforts of different research communities can be -- in some cases rigorously -- connected. Still, the application of this general framework to particular experimental realisations requires the identification of the regimes where the universality phenomenon is expected to appear. In the present review we summarise the recent investigations of many-body quantum systems with long-range interactions, which are currently realised in Rydberg atom arrays, dipolar systems, trapped ion setups and cold atoms in cavity experiments. Our main aim is to present and identify the common and (mostly) universal features induced by long-range interactions in the behaviour of quantum many-body systems. We will discuss both the case of very strong non-local couplings, i.e. the non-additive regime, and the one in which energy is extensive, but nevertheless low-energy, long wavelength properties are altered with respect to the short-range limit. Cases of competition with other local effects in the above mentioned setups are also reviewed.
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