The symmetry of a discrete structure can easily be altered (or in the case of maps, destroyed) by changing just a small part of it. But there are other contexts where a large degree of symmetry is desirable without achieving the maximum possible symmetry, for example with maps and polytopes that are maximally symmetric by orientation-preserving automorphisms but are chiral (admitting no refections). In a similar vein, sometimes restricting the symmetry can help achieve things that are either impossible or difficult when full symmetry is assumed. I will outline a few recent discoveries that explain and illustrate these phenomena.