The symmetry of a discrete object (such as a graph, map or polytope, and even a Riemann surface or 3-manifold) can be measured by its automorphism group – the group of all structure-preserving bijections from the object to itself. (For example, an automorphism of a map takes vertices to vertices, edges to edges, faces to faces, and preserves incidence among these elements.) But also/conversely, objects with specified symmetry can often be constructed from groups.
In this talk I will give some examples showing how this is possible, and some methods which help, and describe some significant outcomes of this approach across a range of topics. Included will be some very recent developments, including a few unexpected discoveries.