  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
$\operatorname{mod}(\mathcal{C})$.
