  Several examples of non-compact manifolds $M_0$ whose geometry at infinity is
described by Lie algebras of vector fields $V \subset \Gamma(TM)$ (on a
compactification of $M_0$ to a manifold with corners $M$) were studied by
Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the
geometry of manifolds described by Lie algebras of vector fields -- baptised
"manifolds with a Lie structure at infinity" there -- was studied from an
axiomatic point of view. In this paper, we define and study the algebra
$\Psi_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential
operators canonically associated to a manifold $M_0$ with the Lie structure at
infinity $V \subset\Gamma(TM)$. We show that many of the properties of the
usual algebra of pseudodifferential operators on a compact manifold extend to
$\Psi_{1,0,V}^\infty(M_0)$. We also consider the algebra $\DiffV{*}(M_0)$ of
differential operators on $M_0$ generated by $V$ and $\CI(M)$, and show that
$\Psi_{1,0,V}^\infty(M_0)$ is a ``microlocalization'' of $\DiffV{*}(M_0)$.
Finally, we introduce and study semi-classical and ``suspended'' versions of
the algebra $\Psi_{1,0,V}^\infty(M_0)$. Our construction solves a problem posed
by Melrose in his talk at the ICM in Kyoto.
