Explore a 55-minute mathematics lecture that surveys Riemannian geometry of diffeomorphism groups and their various metric types, particularly focusing on applications in continuum mechanics and shape analysis. Delve into the historical development of diffeomorphism group curvature studies since Vladimir Arnold's 1996 work, examining how comparison theory determines stability and well-posedness, while also investigating singular behaviors including conjugate points and vanishing geodesic distance. Learn about six decades of mathematical developments from multiple authors in this field, presented as part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" at the Erwin Schrödinger International Institute for Mathematics and Physics.