  We present a global analysis of the geometries that arise in non-compact
current algebra (or gauged WZW) coset models of strings and particles
propagating in curved space-time. The simplest case is the 2d black hole. In
higher dimensions these geometries describe new and much more complex
singularities. For string and particle theories (defined in the text) we
introduce general methods for identifying global coordinates and give the
general exact solution for the geodesics for any gauged WZW model for any
number of dimensions. We then specialize to the 3d geometries associated with
$SO(2,2)/SO(2,1)$ (and also $SO(3,1)/SO(2,1)$) and discuss in detail the global
space, geodesics, curvature singularities and duality properties of this space.
The large-small (or mirror) type duality property is reformulated as an
inversion in group parameter space. The 3d global space has two topologically
distinct sectors, with patches of different sectors related by duality. The
first sector has a singularity surface with the topology of ``pinched double
trousers". It can be pictured as the world sheet of two closed strings that
join into a single closed string and then split into two closed strings, but
with a pinch in each leg of the trousers. The second sector has a singularity
surface with the topology of ``double saddle", pictured as the world sheets of
two infinite open strings that come close but do not touch. We discuss the
geodesicaly complete spaces on each side of these surfaces and interpret the
motion of particles in physical terms. A cosmological interpretation is
suggested and comments are mode on possible physical applications.
