Abstract: Many geometric applications of PDEs depend on transversality results, e.g. in gauge theory or symplectic topology. One needs certain moduli spaces of solutions to elliptic PDEs such as Seiberg-Witten monopoles or pseudoholomorphic curves to be "transversely cut out", so that they are smooth and have the expected dimension. A problem arises: these PDEs often come with natural symmetries, and everyone knows that you cannot have transversality and symmetry at the same time. In this talk, I will outline a general strategy to deal with transversality without sacrificing symmetry, and discuss what conditions need to be checked before you can apply it to your favorite PDE.