I will give a survey of results on the Riemannian geometry of diffeomorphism groups under various types of metrics, which arise either in continuum mechanics such as fluids or in shape analysis such as LDDMM. Since Vladimir Arnold in 1996, the curvature of diffeomorphism groups has been considered in order to use comparison theory to determine stability and well-posedness, along with singular behavior such as conjugate points and vanishing geodesic distance. This talk will be an overview of results due to many authors, covering topics from the past 60 years since Arnold.