  Let $X$ be a compact manifold with boundary. Suppose that the boundary is
fibred, $\phi:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary
defining function. This data fixes the space of `fibred cusp' vector fields,
consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which
are tangent to the fibres of $\phi;$ it is a Lie algebra and $\CI(X)$ module.
This Lie algebra is quantized to the `small calculus' of pseudodifferential
operators $\PsiF*(X).$ 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.
