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Quantum field theory in curved spacetime is a theory wherein matter istreated fully in accord with the principles of quantum field theory, butgravity is treated classically in accord with general relativity. It is notexpected to be an exact theory of nature, but it should provide a goodapproximate description when the quantum effects of gravity itself do not playa dominant role. A major impetus to the theory was provided by Hawking'scalculation of particle creation by black holes, showing that black holesradiate as perfect black bodies. During the past 30 years, considerableprogress has been made in giving a mathematically rigorous formulation ofquantum field theory in curved spacetime. Major issues of principle with regardto the formulation of the theory arise from the lack of Poincare symmetry andthe 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 beovercome for free (i.e., non-self-interacting) quantum fields by formulatingthe theory via the algebraic approach and focusing attention on the local fieldobservables rather than a notion of ``particles''. However, these ideas, bythemselves, were not adequate for the formulation of interacting quantum fieldtheory, even at a perturbative level, since standard renormalizationprescriptions in Minkowski spacetime rely heavily on Poincare invariance andthe existence of a Poincare invariant vacuum state. However, during the pastdecade, great progress has been made, mainly due to the importation into thetheory of the methods of ``microlocal analysis''. This article will describethe historical development of the subject and describe some of the recentprogress.

The History and Present Status of Quantum Field Theory in Curved Spacetime