  Cubature formulas and geometrical designs are described in terms of
reproducing kernels for Hilbert spaces of functions on the one hand, and Markov
operators associated to orthogonal group representations on the other hand. In
this way, several known results for spheres in Euclidean spaces, involving
cubature formulas for polynomial functions and spherical designs, are shown to
generalize to large classes of finite measure spaces $(\Omega,\sigma)$ and
appropriate spaces of functions inside $L^2(\Omega,\sigma)$. The last section
points out how spherical designs are related to a class of reflection groups
which are (in general dense) subgroups of orthogonal groups.
