  Let $\mathscr{C}$ be a $(d+2)$-angulated category with $d$-suspension functor
$\Sigma^d$. Our main results show that every Serre functor on $\mathscr{C}$ is
a $(d+2)$-angulated functor. We also show that $\mathscr{C}$ has a Serre
functor $\mathbb{S}$ if and only if $\mathscr{C}$ has Auslander--Reiten
$(d+2)$-angles. Moreover, $\tau_d=\mathbb{S}\Sigma^{-d}$ where $\tau_d$ is
$d$-Auslander-Reiten translation. These results generalize work by
Bondal-Kapranov and Reiten-Van den Bergh. As an application, we prove that for
a strongly functorially finite subcategory $\mathscr{X}$ of $\mathscr{C}$, the
quotient category $\mathscr{C}/\mathscr{X}$ is a $(d+2)$-angulated category if
and only if $(\mathscr{C},\mathscr{C})$ is an $\mathscr{X}$-mutation pair, and
if and only if $\tau_d\mathscr{X}=\mathscr{X}$.
