This paper provides upper and lower bounds on the kissing number of congruent radius r > 0 spheres in hyperbolic H-n and spherical S-n spaces, for n >= 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) similar to (n - 1) . d(n-1) . B(n-1/2, 1/2) . e((n-1)r), where d(n) is the sphere packing density in Rn, 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.