A graph G = (V, E) is called a bitopological graph if there exist a set X and a set-indexer f on G such that both f(V ) and f  (E) ∪  are topologies on X. The corresponding set-indexer is called a bitopological set-indexer of G. We prove the existence of bitopological set-indexer. We give a characterization of bitopological complete graphs. We define equi-bitopological graphs and establish certain results on equi-bitopological graphs. We identify certain classes of graphs, which are bitopological and define bitopological index β τ (G) of a finite graph G as the minimum cardinality of the underlying set X. We discuss about embeddingand NP-completeness problems of some classes of non-bitopological graphs.