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