Supersymmetry is a symmetry that maintains Bose-Fermi degeneracy, so the spectrum of bosons and fermions has to be identical. Supersymmetry also imposes severe constraints on the allowed interactions between the various particle species. These models are special, and lead to powerful tools that are applicable even for strongly coupled theories. Traditionally, the power of supersymmetry was mostly limited to studying the vacuum structure (more precisely, the chiral ring). Recently, it was realized that supersymmetry leads to remarkable simplifications also on curved spaces, and it allows to extract information that extends well beyond the chiral ring of the theory. These recent developments can also be used to evaluate the expectation values of non-local observables, such as the entanglement entropy of the vacuum. These observables are of exceeding interest in particle physics and also in condensed matter physics. Which additional quantities can be evaluated non-perturbatively in supersymmetric field theories? What can we learn about general aspects of quantum field theory such as dualities and anomalies? What do we learn about the vacuum of quantumfield theory? What is the mathematical interpretation of supersymmetric theories on curved spaces?