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 Φ-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 Poincaré duality between the K-homology of the stratified pseudomanifold and the K-group of fully elliptic operators. Contents 10 3. The definition of S-pseudodifferential operators 13 4. Groupoids 17 5. Action of S-pseudodifferential operators 21 6. Suspended S-operators 24 7. Symbol Maps 28 8. Composition 31 9. Mapping properties 36 10. The semiclassical S-calculus 46 11. Poincaré duality 53 References 62

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