  For each positive integer $n$ we introduce the notion of $n$-exangulated
categories as higher dimensional analogues of extriangulated categories defined
by Nakaoka-Palu. We characterize which $n$-exangulated categories are $n$-exact
in the sense of Jasso and which are $(n+2)$-angulated in the sense of
Geiss-Keller-Oppermann. For extriangulated categories with enough projectives
and injectives we introduce the notion of $n$-cluster tilting subcategories and
show that under certain conditions such $n$-cluster tilting subcategories are
$n$-exangulated.
