The angular bispectrum of spherical random fields has recently gained an
enormous importance, especially in connection with statistical inference on cosmological
data. In this paper, we analyze its moments and cumulants of arbitrary order and
we use these results to establish a multivariate central limit theorem and higher order
approximations. The results rely upon combinatorial methods from graph theory and
a detailed investigation for the asymptotic behavior of coefficients arising in matrix
representation theory for the group of rotations SO(3).