We define and study an algebra Ψ∞1,0,(M0) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields on a compactification M of M0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ∞1,0,(M0). We also consider the algebra Diff∗v(M0) of differential operators on M0 generated by and ∞(M), and show that Ψ∞1,0,(M0) is a microlocalization of Diff∗(M0). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and “suspended” versions of the algebra Ψ∞1,0,(M0).