  We review the definition of a Lie manifold $(M, \VV)$ and the construction of
the algebra $\Psi\sp{\infty}\sb{\VV}(M)$ of pseudodifferential operators on a
Lie manifold $(M, \VV)$. We give some concrete Fredholmness conditions for
pseudodifferential operators in $\Psi\sp{\infty}\sb{\VV}(M)$ for a large class
of Lie manifolds $(M, \VV)$. These Fredholmness conditions have applications to
boundary value problems on polyhedral domains and to non-linear PDEs on
non-compact manifolds. As an application, we determine the spectrum of the
Dirac operator on a manifold with multi-cylindrical ends.
