  This paper is concerned with pseudodifferential calculus on manifolds with
fibred corners. Following work of Connes, Monthubert, Skandalis and
Androulidakis, we associate to every manifold with fibred corners a
longitudinally smooth groupoid which algebraic and differential structure is
explicitely described. This groupoid has a natural geometric meaning as a
holonomy groupoid of singular foliation, it is a singular leaf space in the
sense of Androulidakis and Skandalis. We then show that the associated
compactly supported pseudodifferential calculus coincides with Mazzeo and
Melrose's phi-calculus and we introduce an extended algebra of smoothing
operators that is shown to be stable under holomorphic functional calculus.
This result allows the interpretation of phi-calculus as the pseudodifferential
calculus associated with the holonomy groupoid of the singular foliation
defined by the manifold with fibred corners. It is a key step to set the index
theory of those singular manifolds in the noncommutative geometry framework.
