Digital platforms are shaping the digital age. Recently, the so-called GAFAM (Google, Apple, Facebook, Amazon and Microsoft) as well as other digital platforms have increasingly become the focus of competition authorities. The market power of the respective digital platform is usually the linchpin of the antitrust assessment. However, it is not easy to grasp the market power of digital platforms with the traditional analytical tools of antitrust law. In this contribution, Vicky Robertson shows which difficulties arise when trying to delineate the relevant market(s) in which digital platforms are active. It also discusses how the constituent element of a dominant position can be applied to digital platforms. As an alternative manifestation of market power, relative market power is then examined, as it can be found in both Austria and Germany – but not at the level of the European. Then, a foray through different European legal systems shows that the concept of market power seems to be subject to change in the digital age: legislators in Europe are currently breaking new ground in order to allow competition law to grasp the manifestations of digital market power by digital platforms. The contribution is in German and can be accessed here (pre-edited), and here (behind a paywall).