  We study realizations of pseudodifferential operators acting on sections of
vector-bundles on a smooth, compact manifold with boundary, subject to
conditions of Atiyah-Patodi-Singer type. Ellipticity and Fredholm property,
compositions, adjoints and self-adjointness of such realizations are discussed.
We construct regular spectral triples $(\mathcal{A},\mathcal{H},\mathcal{D})$
for manifolds with boundary of arbitrary dimension, where $\mathcal{H}$ is the
space of square integrable sections. Starting out from Dirac operators with
APS-conditions, these triples are even in case of even dimensional manifolds;
we show that the closure of $\mathcal{A}$ in $\mathscr{L}(\mathcal{H})$
coincides with the continuous functions on the manifold being constant on each
connected component of the boundary.
