Conformal compactification, as originally defined by Penrose, has long been recognised as an effective geometric framework for relating conformal geometry, and associated field theories ``at infinity'', to the asymptotic phenomena of an interior (pseudo-)-Riemannian geometry of one higher dimension.  It provides an effective approach for analytic problems in GR, geometric scattering, conformal invariant theory, as well as the AdS/CFT correspondence of Physics. For many of these applications it should be profitable to consider other notions of geometric compactification. For manifolds $M$ with a complete affine connection $\nabla$, I will define a class of compactifications based around projective geometry (that is the geodesic path structure of $\nabla$). This applies to pseudo-Riemannian geometry via the Levi-Civita connection and provides an effective alternative to conformal compactification. The construction is linked to the solutions of overdetermined PDE known as BGG equations and via this is seen to a part of a very general picture.  This is joint work with Andreas Cap.