The concept of symmetry has received significant attention in computer graphics and computer vision research in recent years. Numerous methods have been proposed to find, extract, encode and exploit geometric symmetries and high-level structural information for a wide variety of geometry processing tasks. This report surveys and classifies recent developments in symmetry detection. We focus on elucidating the key similarities and differences between existing methods to gain a better understanding of a fundamental problem in digital geometry processing and shape understanding in general. We discuss a variety of applications in computer graphics and geometry processing that benefit from symmetry information for more effective processing. An analysis of the strengths and limitations of existing algorithms highlights the plenitude of opportunities for future research both in terms of theory and applications.