  This paper reviews the progress made over the last five years in studying
boundary conditions and semiclassical properties of quantum fields about
4-real-dimensional Riemannian backgrounds. For massless spin-${1\over 2}$
fields one has a choice of spectral or supersymmetric boundary conditions, and
the corresponding conformal anomalies have been evaluated by using
zeta-function regularization. For Euclidean Maxwell theory in vacuum, the
mode-by-mode analysis of BRST-covariant Faddeev-Popov amplitudes has been
performed for relativistic and non-relativistic gauge conditions. For massless
spin-${3\over 2}$ fields, the contribution of physical degrees of freedom to
one-loop amplitudes, and the 2-spinor analysis of Dirac and Rarita-Schwinger
potentials, have been obtained. In linearized gravity, gauge modes and ghost
modes in the de Donder gauge have been studied in detail. This program may lead
to a deeper understanding of different quantization techniques for gauge fields
and gravitation, to a new vision of gauge invariance, and to new points of view
in twistor theory.
