TRAPPIST-1 is a fantastic nearby (∼39.14 light years) planetary system made of at least seven transiting terrestrial-size, terrestrial-mass planets all receiving a moderate amount of irradiation. To date, this is the most observationally favourable system of potentially habitable planets known to exist. Since the announcement of the discovery of the TRAPPIST-1 planetary system in 2016, a growing number of techniques and approaches have been used and proposed to characterize its true nature. Here we have compiled a state-of-the-art overview of all the observational and theoretical constraints that have been obtained so far using these techniques and approaches. The goal is to get a better understanding of whether or not TRAPPIST-1 planets can have atmospheres, and if so, what they are made of. For this, we surveyed the literature on TRAPPIST-1 about topics as broad as irradiation environment, planet formation and migration, orbital stability, effects of tides and Transit Timing Variations, transit observations, stellar contamination, density measurements, and numerical climate and escape models. Each of these topics adds a brick to our understanding of the likely—or on the contrary unlikely—atmospheres of the seven known planets of the system. We show that (i) Hubble Space Telescope transit observations, (ii) bulk density measurements comparison with H2-rich planets mass-radius relationships, (iii) atmospheric escape modelling, and (iv) gas accretion modelling altogether offer solid evidence against the presence of hydrogen-dominated—cloud-free and cloudy—atmospheres around TRAPPIST-1 planets. This means that the planets are likely to have either (i) a high molecular weight atmosphere or (ii) no atmosphere at all. There are several key challenges ahead to characterize the bulk composition(s) of the atmospheres (if present) of TRAPPIST-1 planets. The main one so far is characterizing and correcting for the effects of stellar contamination. Fortunately, a new wave of observations with the James Webb Space Telescope and near-infrared high-resolution ground-based spectrographs on existing very large and forthcoming extremely large telescopes will bring significant advances in the coming decade.