  In this work, we present an overview of uniqueness results derived in recent
years for the quantization of Gowdy cosmological models and for (test)
Klein-Gordon fields minimally coupled to
Friedmann-Lema\^{\i}tre-Robertson-Walker, de Sitter, and Bianchi I spacetimes.
These results are attained by imposing the criteria of symmetry invariance and
of unitary implementability of the dynamics. This powerful combination of
criteria allows not only to address the ambiguity in the representation of the
canonical commutation relations, but also to single out a preferred set of
fundamental variables. For the sake of clarity and completeness in the
presentation (essentially as a background and complementary material), we first
review the classical and quantum theories of a scalar field in globally
hyperbolic spacetimes. Special emphasis is made on complex structures and the
unitary implementability of symplectic transformations.
