These lectures will first describe some uses of group theory in combinatorics, especially to analyse and construct discrete objects with specified symmetry, and then describe some combinatorial methods useful in group theory, especially to analyse and construct subgroups of finite index in finitely-presented groups, and finite quotients of finitely-presented groups. Examples include coset graphs and their transversals. A common theme will be graphs and maps (2-cell embeddings of graphs on surfaces) of particular types, with automorphism groups of largest possible order.