  A proper understanding of boundary-value problems is essential in the attempt
of developing a quantum theory of gravity and of the birth of the universe. The
present paper reviews these topics in light of recent developments in spectral
geometry, i.e. heat-kernel asymptotics for the Laplacian in the presence of
Dirichlet, or Robin, or mixed boundary conditions; completely gauge-invariant
boundary conditions in Euclidean quantum gravity; local vs. non-local
boundary-value problems in one-loop Euclidean quantum theory via path
integrals.
