The classical approach to maps, as surveyed by Coxeter and Moser (“ Generators and Relations for Discrete Groups,” Springer-Verlag, 1980), is by cell decomposition of a surface. A more recent approach, by way of graph embedding schemes, is taken by Edmonds ( Notices Amer. Math. Soc. 7 (1960) , 646), Tutte ( Canad. J. Math. 31 (5) (1979) , 986–1004), and others. Our intention is to formulate a purely combinatorial generalization of a map, called a combinatorial map. Besides maps on orientable and nonorientable surfaces, combinatorial maps include polytopes, tessellations, the hypermaps of Walsh, higher dimensional analogues of maps, and certain toroidal complexes of Coxeter and Shephard ( J. Combin. Theory Ser. B. 22 (1977) , 131–138) and Grünbaum (Colloques internationaux C.N.R.S. No. 260, Problèmes Combinatoire et Théorie des Graphes, Orsay, 1976). The concept of a combinatorial map is formulated graph theoretically. The present paper treats the incidence structure, the diagram, reduciblity, order, geometric realizations, and group theoretic and topological properties of combinatorial maps. Another paper investigates highly symmetric combinatorial maps.