  A Calder\'on projector for an elliptic operator $P$ on a manifold with
boundary $X$ is a projection from general boundary data to the set of boundary
data of solutions $u$ of $Pu=0$. Seeley proved in 1966 that for compact $X$ and
for $P$ uniformly elliptic up to the boundary there is a Calder\'on projector
which is a pseudodifferential operator on $\partial X$. We generalize this
result to the setting of fibred cusp operators, a class of elliptic operators
on certain non-compact manifolds having a special fibred structure at infinity.
This applies, for example, to the Laplacian on certain locally symmetric spaces
or on particular singular spaces, such as a domain with cusp singularity or the
complement of two touching smooth strictly convex domains in Euclidean space.
Our main technical tool is the $\phi$-pseudodifferential calculus introduced by
Mazzeo and Melrose.
  In our presentation we provide a setting that may be useful for doing
analogous constructions for other types of singularities.
