  The 2023 second edition of a 2006 encyclopedia article on mathematical
aspects of quantum field theory in curved spacetimes (QFTCST). Section-titles
(with new sections indicated with stars) are: Introduction and preliminaries,
Construction of a $*$-algebra for a real linear scalar field on globally
hyperbolic spacetimes and some general theorems, *More about (quasifree)
Hadamard states, Particle creation and the limitations of the particle concept,
Theory of the stress-energy tensor, *More about the intersection of QFTCST with
AQFT and the Fewster-Verch No-Go Theorem, Hawking and Unruh effects, *More
about (classical and) quantum fields on black hole backgrounds, Non-globally
hyperbolic spacetimes and the time-machine question, *More about QFT on
non-globally hyperbolic spacetimes, Other related topics and some warnings. The
article contains many references. It also includes a review of, and also
compares and contrasts, recent results on the implications of QFTCST for the
question of the instability of three sorts of Cauchy horizon -- first those
inside black holes such as especially Reissner-Nordstr\"om-de Sitter and
Kerr-de Sitter, second the compactly generated Cauchy horizons of spacetimes in
which time-machines get manufactured, and third the Cauchy horizon of the
spacetime which is believed to describe evaporating black holes and which
underlies (one version of) the black hole information-loss puzzle.
