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Abstract: After discussing what it means to derive the AdS/CFT correspondence, I will describe our attempts (in collaboration with Shai Chester and Erez Urbach, 2011.06328) to make this correspondence explicit in the simplest case of the U(N) vector model. We explicitly rewrite the path integral for the free or critical U(N) (or O(N)) bosonic vector models in d space-time dimensions as a path integral over fields (including massless high-spin fields) living on (d+1)-dimensional anti-de Sitter space. Inspired by de Mello Koch, Jevicki, Suzuki and Yoon and earlier work, we first rewrite the vector models in terms of bi-local fields, then expand these fields in eigenmodes of the conformal group, and finally map these eigenmodes to those of fields on anti-de Sitter space. Our results provide an explicit (non-local) action for a high-spin theory on anti-de Sitter space, which is presumably equivalent in the large N limit to Vasiliev's classical high-spin gravity theory (with some specific gauge-fixing to a fixed background), but which can be used also for loop computations. Our mapping is explicit within the 1/N expansion, but in principle can be extended also to finite N theories, where extra constraints on products of bulk fields need to be taken into account.