ABSTRACT: This paper introduces three new types of combinatorial structures associated with group actions, namely symmetrical covers, symmetrical decompositions, and symmetrical factorisations of graphs. These structures are related to and generalise various combinatorial objects, such as 2-designs, regular maps, near-polygonal graphs, and linear spaces. General theory is developed for each of these structures, pertinent examples and constructions are given, and a number of open research problems are posed.