The entropy of a graph was first introduced by Rashevsky \cite{Rashevsky} and Trucco \cite{Trucco} to interpret as the structural information content of the graph and serve as a complexity measure. In this paper, we first state a number of definitions of graph entropy measures and generalized graph entropies. Then we survey the known results about them from the following three respects: inequalities and extremal properties on graph entropies, relationships between graph structures, graph energies, topological indices and generalized graph entropies, complexity for calculation of graph entropies. Various applications of graph entropies together with some open problems and conjectures are also presented for further research.