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Abstract. We define and study an algebra Ψ∞1,0,V (M0) of pseudodifferential operators canonically associated to a non-compact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields V 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,V (M0). We also consider the algebra Diff∗V (M0) of differential operators on M0 generated by V and C∞(M), and show that Ψ∞1,0,V (M0) is a microlocalization of Diff V (M0). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and stud