  We investigate Stein-Malliavin approximations for nonlinear functionals of
geometric interest of Gaussian random eigenfunctions on the unit $d$
-dimensional sphere ${\mathbb{S}}^{d},$ $d\geq 2.$ All our results are
established in the high energy limit, i.e. for eigenfunctions corresponding to
growing eigenvalues. More precisely, we provide an asymptotic analysis for the
variance of random eigenfunctions, and also establish rates of convergence for
various probability metrics for Hermite subordinated processes, arbitrary
polynomials of finite order and square integral nonlinear transforms; the
latter, for instance, allows to prove a quantitative Central Limit Theorem for
the excursion area. Some related issues were already considered in the
literature for the $2$-dimensional case ${\mathbb{S}}^{2}$; our results are new
or improve the existing bounds even for this special case. Proofs are based on
the asymptotic analysis of moments of all order for Gegenbauer polynomials, and
make extensive use of the recent literature on so-called fourth-moment theorems
by Nourdin and Peccati.
