This paper provides upper and lower bounds on the kissing number of congruent radius
r
>
0
r > 0
spheres in hyperbolic
H
n
\mathbb {H}^n
and spherical
S
n
\mathbb {S}^n
spaces, for
n
≥
2
n\geq 2
. For that purpose, the kissing number is replaced by the kissing function
κ
H
(
n
,
r
)
\kappa _H(n, r)
, resp.
κ
S
(
n
,
r
)
\kappa _S(n, r)
, which depends on the dimension
n
n
and the radius
r
r
.
After we obtain some theoretical upper and lower bounds for
κ
H
(
n
,
r
)
\kappa _H(n, r)
, we study their asymptotic behaviour and show, in particular, that
κ
H
(
n
,
r
)
∼
(
n
−
1
)
⋅
d
n
−
1
⋅
B
(
n
−
1
2
,
1
2
)
⋅
e
(
n
−
1
)
r
\kappa _H(n,r) \sim (n-1) \cdot d_{n-1} \cdot B(\frac {n-1}{2}, \frac {1}{2}) \cdot e^{(n-1) r}
, where
d
n
d_n
is the sphere packing density in
R
n
\mathbb {R}^n
, and
B
B
is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of
κ
S
(
n
,
r
)
\kappa _S(n, r)
, for
n
=
3
,
4
n= 3,\, 4
, over subintervals in
[
0
,
π
]
[0, \pi ]
with relatively high accuracy.