  On conformally compact manifolds of arbitrary signature, we use conformal
geometry to identify a natural (and very general) class of canonical boundary
problems. It turns out that these encompass and extend aspects of already known
holographic bulk-boundary problems, the conformal scattering description of
boundary conformal invariants, and corresponding questions surrounding a range
of physical bulk wave equations. These problems are then simultaneously solved
asymptotically to all orders by a single universal calculus of operators that
yields what may be described as a solution generating algebra. The operators
involved are canonically determined by the bulk (i.e. interior) conformal
structure along with a field which captures the singular scale of the boundary;
in particular the calculus is canonical to the structure and involves no
coordinate choices. The generic solutions are also produced without recourse to
coordinate or other choices, and in all cases we obtain explicit universal
formulae for the solutions that apply in all signatures and to a range of
fields. A specialisation of this calculus yields holographic formulae for GJMS
operators and Branson's Q-curvature.
