We axiomatise the theory of
(
∞
,
n
)
(\infty ,n)
-categories. We prove that the space of theories of
(
∞
,
n
)
(\infty ,n)
-categories is a
B
(
Z
/
2
)
n
B(\mathbb {Z}/2)^n
. We prove that Rezk’s complete Segal
Θ
n
\Theta _n
spaces, Simpson and Tamsamani’s Segal
n
n
-categories, the first author’s
n
n
-fold complete Segal spaces, Kan and the first author’s
n
n
-relative categories, and complete Segal space objects in any model of
(
∞
,
n
−
1
)
(\infty , n-1)
-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of
(
Z
/
2
)
n
(\mathbb {Z}/2)^n
.