Virtually all of the important IT companies have been under scrutiny from the competition authorities, often on a worldwide basis. However, this is not yet true for the main actors in the field of social media. This article explores the competition law concerns of this sector focusing on the most popular platform, i.e., Facebook, and its potential abuses of dominant position under Article 102 TFEU. In the world of social media markets, market definition is complicated because of the two-sided nature of the platforms. Hence, a new version of the SSNIP test is proposed here which rejects the application of a price-based test to free Internet services and re-evaluates the importance of quality as opposed to price. Network effects and other barriers to entry are traditionally discussed for the establishment of dominance. Prohibited behaviour may be either exploitative or exclusionary. Potential exploitative conduct is reviewed from the perspective of private social media users, which highlights the controversial attitude of Facebook towards changes in administration and design by the platform, protection of information and deletion of profiles. Past IT-related case law is still relevant for the identification of exclusionary behaviour, especially as regards the rules on tying, bundling and leveraging. Developing data protection law influences restrictions on data portability. Overall, the hopeful conclusion tends to be that competition law misgivings could be allayed by a suitable adaptation of the general regulatory framework to avoid abuse in this rapidly growing area.