ABSTRACT: This paper gives a brief summary of various aspects of combinatorial group theory and associated computational methods, with special reference to finitely-presented groups and their applications, found useful in the study of graphs, maps and polytopes having maximal symmetry. Recent results include the determination of all arc-transitive cubic graphs on up to 2048 vertices, and of all regular maps of genus 2 to 100, and construction of the first known examples of finite chiral 5-polytopes. Moreover, patterns in the maps data have led to new theorems about the genus spectrum of chiral maps and regular maps with simple underlying graph.