  We study the geometry of hyperbolic cone surfaces, possibly with cusps or
geodesic boundaries. We prove that any hyperbolic cone structure on a surface
of non-exceptional type is determined up to isotopy by the geodesic lengths of
a finite specific homotopy classes of non-peripheral simple closed curves. As
an application, we show that the Thurston asymmetric metric is well-defined on
the Teichm\"uller space of hyperbolic cone surfaces with fixed cone angles and
boundary lengths. We compare such a Teichm\"uller space with the Teichm\"uller
space of complete hyperbolic surfaces with punctures, by showing that the two
spaces (endowed with the Thurston metric) are almost isometric.
