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A lot of efforts have been devoted in the last decade to the investigation ofthe high-frequency behaviour of geometric functionals for the excursion sets ofrandom spherical harmonics, i.e., Gaussian eigenfunctions for the sphericalLaplacian $\Delta_{\mathbf{S}^2}$. In this survey we shall review some of theseresults, with particular reference to the asymptotic behaviour of variances,phase transitions in the nodal case (the \emph{Berry's CancellationPhenomenon}), the distribution of the fluctuations around the expected values,and the asymptotic correlation among different functionals. We shall alsodiscuss some connections with the Gaussian Kinematic Formula, with Wiener-Chaosexpansions and with recent developments in the derivation of QuantitativeCentral Limit Theorems (the so-called Stein-Malliavin approach).

Some Recent Developments on the Geometry of Random Spherical Eigenfunctions