The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $mathcal{E}$ are reflected in $operatorname{mod_{mathsf{adm}}}(mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ as a subcategory of $operatorname{mod}(mathcal{E})$ when $mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $mathcal{C}$ and certain resolving subcategories of $operatorname{mod}(mathcal{C})$.
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