  We review what is known about boundary conditions in General Relativity on a
spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in
which one specifies the boundary geometry, is actually not elliptic and in
general does not lead to a well-defined perturbation theory. It is
better-behaved if the extrinsic curvature of the boundary is suitably
constrained, for instance if it is positive- or negative-definite. A different
boundary condition, in which one specifies the conformal geometry of the
boundary and the trace of the extrinsic curvature, is elliptic and always leads
formally to a satisfactory perturbation theory. These facts might have
interesting implications for semiclassical approaches to quantum gravity.
(Submitted to a volume in honor of Roman Jackiw.)
