We prove an index theorem for families of elliptic boundary value problems and a
glueing formula for the index of a family of Dirac operators on a closed manifold. In
the process we also obtain a very general result about the cobordism invariance of
the index of a family.
To achieve these goals we develop techniques (inspired from symplectic geometry)
for computing the index of a family of Fredholm operators.
Key words and phrases: Dirac operators, boundary value problems, Calderon
projectors, index of families, Clifford algebras, Karoubi's A'p'9-theory, generalized
symplectic spaces, generalized symplectic reduction, generalized Maslov index.