  This paper is devoted to mathematical and physical properties of the Dirac
operator and spectral geometry. Spin-structures in Lorentzian and Riemannian
manifolds, and the global theory of the Dirac operator, are first analyzed.
Elliptic boundary-value problems, index problems for closed manifolds and for
manifolds with boundary, Bott periodicity and K-theory are then presented. This
makes it clear why the Dirac operator is the most fundamental, in the theory of
elliptic operators on manifolds. The topic of spectral geometry is developed by
studying non-local boundary conditions of the Atiyah-Patodi-Singer type, and
heat-kernel asymptotics for operators of Laplace type on manifolds with
boundary. The emphasis is put on the functorial method, which studies the
behaviour of differential operators, boundary operators and heat-kernel
coefficients under conformal rescalings of the background metric. In the second
part, a number of relevant physical applications are studied: non-local
boundary conditions for massless spin-1/2 fields, massless spin-3/2 potentials
on manifolds with boundary, geometric theory of massive spin-3/2 potentials,
local boundary conditions in quantum supergravity, quark boundary conditions,
one-loop quantum cosmology, conformally covariant operators and Euclidean
quantum gravity.
