We generalize Contou-Carrère symbols to higher dimensions. To an
(
n
+
1
)
(n+1)
-tuple
f
0
,
…
,
f
n
∈
A
(
(
t
1
)
)
⋯
(
(
t
n
)
)
×
f_0,\dots ,f_n \in A((t_1))\cdots ((t_n))^{\times }
, where
A
A
denotes an algebra over a field
k
k
, we associate an element
(
f
0
,
…
,
f
n
)
∈
A
×
(f_0,\dots ,f_n) \in A^{\times }
, extending the higher tame symbol for
k
=
A
k = A
, and earlier constructions for
n
=
1
n = 1
by Contou-Carrère, and
n
=
2
n = 2
by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic
K
K
-theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols.