The Digital Markets Act (DMA) is a new instrument of EU competition policy. It has been pointed out that although the DMA does not include any provisions on private enforcement, there should still be room for private enforcement of rights under the DMA. However, it has not been properly scrutinised to what extent the provisions of the DMA are suitable for such private enforcement. This article contributes to the literature by addressing this deficit and explaining the interwoven aspects of EU law and national law upon which private enforcement of the DMA will rely. The analysis is carried out in two main tiers. First, the core substantive provisions of the DMA are analysed. It is concluded in this stage that there are some rules in the DMA which may trigger private law sanctions when breached. Second, the system of judicial protection of rights conferred on individuals under EU law is explained and applied to the DMA. In this regard, it is concluded that it would be appropriate to extend any private law remedies that are available to claimants who have been victims of an infringement of Article 101 or 102 TFEU to claimants who have been victims of an infringement of protective provisions in the DMA.