  We extend the method of layer potentials to manifolds with boundary and
cylindrical ends. To obtain this extension along the classical lines, we have
to deal with several technical difficulties due to the non-compactness of the
boundary, which prevents us from using the standard characterization of
Fredholm and compact (pseudo-)differential operators between Sobolev spaces.
Our approach, which involves the study of layer potentials depending on a
parameter on compact manifolds as an intermediate step, yields the
invertibility of the relevant boundary integral operators in the global,
non-compact setting, which is rather unexpected. As an application, we prove
the well-posedness of the non-homogeneous Dirichlet problem on manifolds with
boundary and cylindrical ends. We also prove the existence of the
Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in
the calculus of pseudodifferential operators that are ``almost translation
invariant at infinity,'' a calculus that is closely related to Melrose's
b-calculus \cite{me81, meaps}, which we study in this paper. The proof of the
convergence of the layer potentials and of the existence of the
Dirichlet-to-Neumann map are based on a good understanding of resolvents of
elliptic operators that are translation invariant at infinity.
