In this talk, I will introduce and motivate Cauchy-Riemann (CR) geometry by considering real hypersurfaces embedded in complex Euclidean space. Firstly, I will discuss progress on both Darboux- and Alexandrov-type theorems in this setting. Secondly, I will introduce flows of CR hypersurfaces that are analogous to the mean curvature flow. Alongside the standard degeneracy due to tangential diffeomorphisms, such flows have an additional degeneracy due to the CR structure which will be discussed. Finally, I will discuss joint research with Ben Andrews on new flows which preserve key components of the CR structure. This talk will be accessible to those with a background in Riemannian geometry.
