  In this paper we present some recent results concerning linear spectral
transformations of Carath\'eodory functions. More precisely, given two
Carath\'eodory functions related by a linear spectral transformation, we study
the relation between the corresponding moment functionals and, in the positive
definite case, the relation between the measures.
  We will see that rational modifications of functionals are included in the
linear spectral transformations. However, we will show that there exist a huge
class of linear spectral transformations which are not given by rational
modifications of functionals. Indeed, we will characterize those linear
spectral transformation which come from a rational modification.
  In the general case we will discuss the relation between the functionals
involved in a linear spectral transformation, which allows us to identify the
difficulties to connect the related functionals.
  Actually, several examples will show how amazing can be the relationships
between the moment functionals associated with a linear spectral
transformation.
