  Let $\mathcal{E}=(\mathcal{A},\mathcal{S})$ be an exact category with enough
projectives $\mathcal{P}$. We introduce the notion of support $\tau$-tilting
subcategories of $\mathcal{E}$. It is compatible with existing definitions of
support $\tau$-tilting modules (subcategories) in various context. It is also a
generalization of tilting subcategories of exact categories. We show that there
is a bijection between support $\tau$-tilting subcategories and certain
$\tau$-cotorsion pairs. Given a support $\tau$-tilting subcategory
$\mathcal{T}$, we find a subcategory $\mathcal{E}_{\mathcal{T}}$ of
$\mathcal{E}$ which is an exact category and $\mathcal{T}$ is a tilting
subcategory of $\mathcal{E}_{\mathcal{T}}$. If $\mathcal{E}$ is Krull-Schmidt,
we prove the cardinal $|\mathcal{T}|$ is equal to the number of isomorphism
classes of indecomposable projectives $Q$ such that ${\rm
Hom}_{\mathcal{E}}(Q,\mathcal{T})\neq 0$. We also show a functorial version of
Brenner-Butler's theorem.
