  A geometric transition is a continuous path of geometric structures that
changes type, meaning that the model geometry, i.e. the homogeneous space on
which the structures are modeled, abruptly changes. In order to rigorously
study transitions, one must define a notion of geometric limit at the level of
homogeneous spaces, describing the basic process by which one homogeneous
geometry may transform into another. We develop a general framework to describe
transitions in the context that both geometries involved are represented as
sub-geometries of a larger ambient geometry. Specializing to the setting of
real projective geometry, we classify the geometric limits of any sub-geometry
whose structure group is a symmetric subgroup of the projective general linear
group. As an application, we classify all limits of three-dimensional
hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and
Sol geometry among the limits. We prove, however, that the other Thurston
geometries, in particular $\mathbb{H}^2 \times \mathbb{R}$ and
$\widetilde{\operatorname{SL}_2 \mathbb{R}}$, do not embed in any limit of
hyperbolic geometry in this sense.
