  In this PhD Thesis we investigate the geometry of random fields on compact
Riemannian manifolds, in particular the two-dimensional sphere. In the first
part, we characterize isotropic Gaussian fields on homogeneous spaces of a
compact group and then we prove the non-existence of P. L\'evy's Brownian field
on the group SO(3), which moreover allows to extend the same kind of result to
SO(n), SU(n) for n bigger than 3. In the second part, we investigate the
high-energy behavior of random eigenfunctions on the (hyper)sphere and on the
torus proving quantitative CLT results for some geometric functionals of those
eiegenfunctions. A nice non-Central and non-Universal result is shown for nodal
length distribution in the case of arithmetic random waves. Finally, we extend
representation formulas obtained in the first part to the case of spin random
fields on the sphere, introducing a new approach i.e., the pullback random
field, that easily allows to study random sections of homogeneous vector
bundles.
