  In these three lectures we will discuss some fundamental aspects of the
theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac
operators on compact Riemannian manifolds with smooth boundary emphasizing the
relation with the theory of global boundary conditions.
  Self-adjoint extensions of symmetric operators, specially of the
Laplace-Beltrami and Dirac operators, are fundamental in Quantum Physics as
they determine either the energy of quantum systems and/or their unitary
evolution. The well-known von Neumann's theory of self-adjoint extensions of
symmetric operators is not always easily applicable to differential operators,
while the description of extensions in terms of boundary conditions constitutes
a more natural approach. Thus an effort is done in offering a description of
self-adjoint extensions in terms of global boundary conditions showing how an
important family of self-adjoint extensions for the Laplace-Beltrami and Dirac
operators are easily describable in this way.
  Moreover boundary conditions play in most cases an significant physical role
and give rise to important physical phenomena like the Casimir effect. The
geometrical and topological structure of the space of global boundary
conditions determining regular self-adjoint extensions for these fundamental
differential operators is described. It is shown that there is a natural
homology class dual of the Maslov class of the space.
  A new feature of the theory that is succinctly presented here is the relation
between topology change on the system and the topology of the space of
self-adjoint extensions of its Hamiltonian. Some examples will be commented and
the one-dimensional case will be thoroughly discussed.
