I will discuss new methods in QFTs which allows us to study the non-perturbative dynamics of (non-)supersymmetric theories. The main tools are mixed anomalies, semi-classical methods, resurgence and quantum distillation. As an application, I will describe the charge-q Schwinger model and two-dimensional sigma models, such as CP^{N-1} model compactified on cylinder. Using certain boundary conditions which guarantee persistence of mixed anomaly upon compactification reveals a large set of new saddle points, which playinstrumental role in non-perturbative dynamics.