  Spin (spherical) random fields are very important in many physical
applications, in particular they play a key role in Cosmology, especially in
connection with the analysis of the Cosmic Microwave Background radiation.
These objects can be viewed as random sections of the s-th complex tensor power
of the tangent bundle of the 2-sphere. In this paper, we discuss how to
characterize their expected geometry and topology. In particular, we
investigate the asymptotic behaviour, under scaling assumptions, of general
classes of geometric and topological functionals including Lipschitz-Killing
Curvatures and Betti numbers for (properly defined) excursion sets; we cover
both the cases of fixed and diverging spin parameters s. In the special case of
monochromatic fields (i.e., spin random eigenfunctions) our results are
particularly explicit; we show how their asymptotic behaviour is non-universal
and we can obtain in particular complex versions of Berry's random waves and of
Bargmann-Fock's models as subcases of a new generalized model, depending on the
rate of divergence of the spin parameter s.
