We analyse the structure of the market for foundation models, i.e., large AI models such as those that power ChatGPT and that are adaptable to downstream uses, and we examine the implications for competition policy and regulation. We observe that the most capable models will have a tendency towards natural monopoly and may have potentially vast markets. This calls for a two-pronged regulatory response: (i) Antitrust authorities need to ensure the contestability of the market by tackling strategic behaviour, in particular by ensuring that monopolies do not propagate vertically to downstream uses, and (ii) given the diminished potential for market discipline, there isa role for regulators to ensure that the most capable models meet sufficient quality standards (including safety, privacy, non-discrimination, reliability and interoperability standards) to maximally contribute to social welfare. Regulators should also ensure a level regulatory playing field between AI and non-AI applications in all sectors of the economy. For models that are behind the frontier, we expect competition to be quite intense, implying a more limited role for competition policy, although a role for regulation remains.