  We present a uniform framework generalising and extending the classical
theories of projective differential geometry, c-projective geometry, and almost
quaternionic geometry. Such geometries, which we call \emph{projective
parabolic geometries}, are abelian parabolic geometries whose flat model is an
R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation
$\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We
also give a classification of projective parabolic geometries with
$H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned
classical geometries, includes a geometry modelled on the Cayley plane
$\mathbb{OP}^2$ and conformal geometries of various signatures.
  The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic
structure of a projective parabolic geometry. In particular, by exploiting a
Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on
the Lie algebra of $H$ in which we have tight control over Lie brackets between
various summands. This allows us to generalise known results from the classical
theories. For example, which riemannian metrics are compatible with the
underlying geometry is controlled by the first BGG operator associated to
$\mathbb{W}$.
  In the final chapter, we describe projective parabolic geometries admitting a
$2$-dimensional family of compatible metrics. This is the usual setting for the
classical projective structures; we find that many results which hold in these
settings carry over with little to no changes in the general case.
