  Quantum field theory in curved spacetime is a theory wherein matter is
treated fully in accord with the principles of quantum field theory, but
gravity is treated classically in accord with general relativity. It is not
expected to be an exact theory of nature, but it should provide a good
approximate description when the quantum effects of gravity itself do not play
a dominant role. A major impetus to the theory was provided by Hawking's
calculation of particle creation by black holes, showing that black holes
radiate as perfect black bodies. During the past 30 years, considerable
progress has been made in giving a mathematically rigorous formulation of
quantum field theory in curved spacetime. Major issues of principle with regard
to the formulation of the theory arise from the lack of Poincare symmetry and
the absence of a preferred vacuum state or preferred notion of ``particles''.
By the mid-1980's, it was understood how all of these difficulties could be
overcome for free (i.e., non-self-interacting) quantum fields by formulating
the theory via the algebraic approach and focusing attention on the local field
observables rather than a notion of ``particles''. However, these ideas, by
themselves, were not adequate for the formulation of interacting quantum field
theory, even at a perturbative level, since standard renormalization
prescriptions in Minkowski spacetime rely heavily on Poincare invariance and
the existence of a Poincare invariant vacuum state. However, during the past
decade, great progress has been made, mainly due to the importation into the
theory of the methods of ``microlocal analysis''. This article will describe
the historical development of the subject and describe some of the recent
progress.
