$n$-abelian and $n$-exact categories

We introduce $n$-abelian and $n$-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that $n$-cluster-tilting subcategories of abelian (resp. exact) categories are $n$-abelian (resp. $n$-exact). These results allow to construct several examples of $n$-abelian and $n$-exact categories. Conversely, we prove that $n$-abelian categories satisfying certain mild assumptions can be realized as $n$-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius $n$-exact category has a natural $(n+2)$-angulated structure in the sense of Gei\ss-Keller-Oppermann. We give several examples of $n$-abelian and $n$-exact categories which have appeared in representation theory, commutative ring theory, commutative and non-commutative algebraic geometry.