  Glickenstein \cite{Glickenstein} and Glickenstein-Thomas \cite{GT} introduced
the discrete conformal structures on surfaces in an axiomatic approach and
studied its classification. In this paper, we give a full classification of the
discrete conformal structures on surfaces, which completes Glickenstein-Thomas'
classification. As a result, we find some new classes of discrete conformal
structures on surfaces, including some of the generalized circle packing
metrics introduced by Guo-Luo \cite{GL2}. The relationships between the
discrete conformal structures on surfaces and the 3-dimensional hyperbolic
geometry are also discussed.
