Many properties of physical systems or Riemannian manifolds are encoded in the spectrum of certain interesting, mostly Laplace-type, differential operators. These properties are analyzed by considering suitable functions of the spectrum. This introduction provides an overview of this book. The book provides a comprehensive overview of developments which allow for the analysis of the spectral functions. It develops and applies techniques to analyze determinants arising when the external conditions originate from boundaries present (Casimir effect), dielectric media, scalar backgrounds and magnetic backgrounds. In the context of the electromagnetic field external conditions are most naturally provided by introducing dielectrics. In fact it has been shown that the force between dielectric slabs can be understood as the response of the vacuum to the presence of the dielectrics. The book also outlines briefly the fields of the applications envisaged within quantum field theory under external conditions.