Abstract: The diffeomorphism group of a smooth manifold is really large, yet it is still possible to make sense of it as an infinite-dimensional Lie group. Things get more complicated when we look at subgroups preserving geometric structures. In this talk we will look at a few things: the smooth structure on the group of diffeomorphisms that makes it into a Lie group; subgroups preserving several geometric structures, such as metrics, complex structures, symplectic structures; how Lie groupoids can be used to describe the symmetries of several "intransitive" geometric structures, such as manifolds with a distinguished hypersurface (e.g. a boundary) and foliations. The main focus is on the geometry, so no technical knowlegde of infinite-dimensional manifolds is required. Also, Lie groupoids will only be used at the very end to provide context to some examples, so no worries if you have never seen them before. Hope to see many of you on Friday!