  In this short survey we recollect some of the recent results on the high
energy behavior (i.e., for diverging sequences of eigenvalues) of nonlinear
functionals of Gaussian eigenfunctions on the $d$-dimensional sphere $\mathbb
S^d$, $d\ge 2$. We present a quantitative Central Limit Theorem for a class of
functionals whose Hermite rank is two, which includes in particular the
empirical measure of excursion sets in the non-nodal case. Concerning the nodal
case, we recall a CLT result for the defect on $\mathbb S^2$. The key tools are
both, the asymptotic analysis of moments of all order for Gegenbauer
polynomials, and so-called Fourth-Moment theorems.
