Mathematical models for incompressible fluids play a central role in oceanography and atmospheric science. Despite their relevance, fundamental questions concerning their stability remain largely open. A major research direction in the analysis of partial differential equations (PDEs) is devoted to addressing these issues, as a rigorous mathematical understanding can both guide and improve computational modeling. In this talk, I will present some recent results on stability and instability phenomena for two dimensional non-homogeneous incompressible fluids, with particular emphasis on mechanisms driving unstable dynamics. Opening Address by Serena Cenatiempo (GSSI).