Compute or computing power refers to a software and hardware stack, such as in a data centre or computer, engineered for AI-specific applications. We argue that the antitrust and regulatory literature to date has failed to pay sufficient attention to compute, despite compute being a key input to AI progress and services, the potentially substantial market power of companies in the supply chain, and the advantages of compute as a ‘unit’ of regulation in terms of detection and remedies. We explore potential topics of interest to competition law under merger control, abuse of dominance, state aid, and cartels and collusion. Major companies and states view the development of AI over the coming decades as core to their interests, due to its profound impact on economies, societies and balance of power. If the rapid pace of AI progress is sustained over the long-term, these impacts could be transformative in scale. This potential market power and policy importance should make compute an area of significant interest to antitrust and other regulators.