  This paper provides upper and lower bounds on the kissing number of congruent
radius $r > 0$ spheres in hyperbolic $\mathbb{H}^n$ and spherical
$\mathbb{S}^n$ spaces, for $n\geq 2$. For that purpose, the kissing number is
replaced by the kissing function $\kappa_H(n, r)$, resp. $\kappa_S(n, r)$,
which depends on the dimension $n$ and the radius $r$. After we obtain some
theoretical upper and lower bounds for $\kappa_H(n, r)$, we study their
asymptotic behaviour and show, in particular, that $\kappa_H(n,r) \sim (n-1)
\cdot d_{n-1} \cdot B(\frac{n-1}{2}, \frac{1}{2}) \cdot e^{(n-1) r}$, where
$d_n$ is the sphere packing density in $\mathbb{R}^n$, and $B$ 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 $\kappa_S(n, r)$,
for $n= 3,\, 4$, over subintervals in $[0, \pi]$ with relatively high accuracy.
