Spectral expansions of cosmological fields

We give a review of the theory of random fields defined on the observable part of the Universe that satisfy the cosmological principle, i.e., invariant with respect to the 6-dimensional group $\mathcal{G}$ of the isometries of the time slice of the Friedmann--Lema\^{\i}tre--Robertson--Walker standard chart. Our new results include proof of spectral expansions of scalar and spin weighted $\mathcal{G}$-invariant cosmological fields in open, flat, and closed cosmological models.