  One way to geometrically encode the singularities of a stratified
pseudomanifold is to endow its interior with an iterated fibred cusp metric.
For such a metric, we develop and study a pseudodifferential calculus
generalizing the \Phi-calculus of Mazzeo and Melrose. Our starting point is the
observation, going back to Melrose, that a stratified pseudomanifold can be
`resolved' into a manifold with fibred corners. This allows us to define
pseudodifferential operators as conormal distributions on a suitably blown-up
double space. Various symbol maps are introduced, leading to the notion of full
ellipticity. This is used to construct refined parametrices and to provide
criteria for the mapping properties of operators such as Fredholmness or
compactness. We also introduce a semiclassical version of the calculus and use
it to establish a Poincar\'e duality between the K-homology of the stratified
pseudomanifold and the K-group of fully elliptic operators.
