  We review the mathematically rigorous formulation of the quantum theory of a
linear field propagating in a globally hyperbolic spacetime. This formulation
is accomplished via the algebraic approach, which, in essence, simultaneously
admits all states in all possible (unitarily inequivalent) Hilbert space
constructions. The physically nonsingular states are restricted by the
requirement that their two-point function satisfy the Hadamard condition, which
insures that the ultra-violet behavior of the state be similar to that of the
vacuum state in Minkowski spacetime, and that the expected stress-energy tensor
in the state be finite. We briefly review the Unruh and Hawking effects from
the perspective of the theoretical framework adopted here. A brief discussion
also is given of several open issues and questions in quantum field theory in
curved spacetime regarding the treatment of ``back-reaction", the validity of
some version of the ``averaged null energy condition'', and the formulation and
properties of quantum field theory in causality violating spacetimes.
