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Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ : ∂X −→ Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of 'fibred cusp' vector fields, consisting of those vector fields V on X satisfying V x = O(x 2 ) and which are tangent to the fibres of φ; it is a Lie algebra and C ∞ (X) module. This Lie algebra is quantized to the 'small calculus' of pseudodifferential operators Ψ * Φ (X). Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an 'exact fibred cusp' metric is analyzed as is the wavefront set associated to the calculus.

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