  In this paper, we review the history, current state-of-art, and physical
applications of the spectral theory of two classes of random functions. One
class consists of homogeneous and isotropic random fields defined on a
Euclidean space and taking values in a real finite-dimensional linear space. In
applications to continuum physics, such a field describes physical properties
of a homogeneous and isotropic continuous medium in the situation, when a
microstructure is attached to all medium points. The range of the field is the
fixed point set of a symmetry class, where two compact Lie groups act by
orthogonal representations. The material symmetry group of a homogeneous medium
is the same at each point and acts trivially, while the group of physical
symmetries may act nontrivially. In an isotropic random medium, the rank 1
(resp. rank 2) correlation tensors of the field transform under the action of
the group of physical symmetries according to the above representation (resp.
its tensor square), making the field isotropic.
  Another class consists of isotropic random cross-sections of homogeneous
vector bundles over a coset space of a compact Lie group. In applications to
cosmology, the coset space models the sky sphere, while the random
cross-section models a cosmic background. The Cosmological Principle ensures
that the cross-section is isotropic.
  For convenience of the reader, a necessary material from multilinear algebra,
representation theory, and differential geometry is reviewed in Appendix.
