  Several situations of physical importance may be modelled by linear quantum
fields propagating in fixed spacetime-dependent classical background fields.
For example, the quantum Dirac field in a strong and/or time-dependent external
electromagnetic field accounts for the creation of electron-positron pairs out
of the vacuum. Also, the theory of linear quantum fields propagating on a given
background curved spacetime is the appropriate framework for the derivation of
black-hole evaporation (Hawking effect) and for studying the question whether
or not it is possible in principle to manufacture a time-machine. It is a
well-established metatheorem that any question concerning such a linear quantum
field may be reduced to a definite question concerning the corresponding
classical field theory (i.e. linear hyperbolic PDE with non-constant
coefficients describing the background in question) -- albeit not necessarily a
question which would have arisen naturally in a purely classical context. The
focus in this talk will be on the covariant Klein-Gordon equation in a fixed
curved background, although we shall draw on analogies with other background
field problems and with the time-dependent harmonic oscillator. The aim is to
give a sketch-impression of the whole subject of Quantum Field Theory in Curved
Spacetime, focussing on work with which the author has been personally
involved, and also to mention some ideas and work-in-progress by the author and
collaborators towards a new "semi-local" vacuum construction for this subject.
A further aim is to introduce, and set into context, some recent advances in
our understanding of the general structure of quantum fields in curved
spacetimes which rely on classical results from microlocal analysis.
