This talk will survey the theory
of locally homogeneous geometric
structures on manifolds. Such a structure
is given by a system of local coordinates
modeled on a ``geometry'' (a homogeneous space of a Lie
group). A familiar example is that the sphere admits
no Euclidean-geometry structure: no metrically
accurate world atlas of the earth exists.
When a geometric structure does exist, they
form a space which itself carries interesting
geometry. This talk will discuss several
types of geometric structures, and will end
with the classification of complete affine
structures on 3-manifolds (joint work with
Charette, Drumm, Fried, Labourie and Margulis).