With data as an important parameter for success in markets, issues of the data economy become relevant for competition law. This field of the law traditionally deals with the functioning of market mechanisms and power of individual firms. Competition authorities and courts have to adapt to paradigm shifts in many areas: the value of data can hardly be monetized. Markets in the digital economy are often multi-sided and shaped by strong network effects. Powerful platform operators may control access for customers. Access to data may become a market-entry barrier. Finally, the use of data to feed algorithms and AI may even change competition as such. We examine the data economy from a competition law perspective and present the relevant theories of harm by means of a sample of leading cases, mainly of European courts and competition authorities. Finally, we touch upon first regulatory responses and related questions.