This paper provides upper and lower bounds on the kissing number of congruent radius r> 0 spheres in hyperbolic Hn and spherical Sn spaces, for n≥ 2. For that purpose, the kissing number is replaced by the kissing function κH(n, r), resp. κS(n, r), which depends on the dimension n and the radius r. After we obtain some theoretical upper and lower bounds for κH(n, r), we study their asymptotic behaviour and show, in particular, that κH(n,r)∼(n-1)·dn-1·B(n-12,12)·e(n-1)r, where dn 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 κS(n, r), for n=3,4, over subintervals in [ 0, π] with relatively high accuracy.