Increasingly, observations, experiments, and numerical simulations reveal a rich nonlinear world full of complex structures. Geometric and topological techniques can provide valuable assistance in making sense of these structures. This chapter describes crossing numbers and helicity integrals, which measure different aspects of topological complexity in vector fields. These measures can be applied to vortex structures in fluid mechanics and to magnetic field line structures in magnetized fluids. We also discuss in detail astrophysical applications concerning magnetic fields in the sun and solar atmosphere.