I give an overview of work with Aasen and Mong on topological defects in two-dimensional classical lattice models, quantum spin chains and tensor networks. The partition function in the presence of a topological defect is invariant under any local deformation of the defect. By using results from fusion categories, we construct topological defects in a wide class of lattice models, and show how to use them to derive exact properties of field theories by explicit lattice calculations. In the Ising model, the fusion of duality defects allows Kramers-Wannier duality to be enacted on the torus and higher genus surfaces easily, implementing modular invariance directly on the lattice. In other models, the construction leads to generalised dualities previously unknown. A consequence is an explicit definition of twisted boundary conditions that yield the precise shift in momentum quantization and for critical theories, the spin of the associated conformal field. Other universal quantities we compute exactly on the lattice are the ratios of g-factors for conformal boundary conditions