Since the establishment of commercial sharing economy services like Uber, Blablacar, Lyft, Airbnb, TaskRabbit, etc., the debate about the sharing economy and its effects on competition has generated lively discussions, which have too often dangerously departed from a debate based on objective (market) observation to evolve into a quarrel among the supporters and opponents of the online platforms. Undoubtedly, the peculiar features of these new firms' business models create frictions with the traditional regulatory environment, which currently appears to be incapable of framing them into models and schemes typical of a previous economic phase, such as, for example, one-sided markets, no externalities, and competition mainly on price. Nevertheless, setting aside the more or less impromptu debate about the "social goodness" of these firms, we argue that competition enforcers should look at their effective market power. In fact, as the basic principles of competition law teach us, only when those firms have (more or less legitimate) significant market power, will they be subject to special responsibilities and to stringent restrictions and obligations. Toward this aim, it is first necessary to define the relevant market. And, immediately afterwards, to delimit firms' market position. This, in turn, should help to assess their compliance with the competition rules and the obligations that they are – or rather that they should be – subjected to. This exercise is not an easy one because the traditional regulatory concepts and definitions do not seem to reflect the competition dynamics that characterise the new markets on which we are reflecting. In this paper we focus on a number of challenges that are posed by the sharing economy businesses, suggesting that they could be solved with the traditional competition instruments, although adapted to the peculiar features of the markets that are at stake. These include, among others, multi-sidedness and the presence of different externalities.