The formalism of higher categories has played an increasingly important role in the study of topological quantum field theories. In this survey paper, the authors introduce the notion of field theories with defects, together with examples. Given a bordism category, the additional data of defect conditions consists of submanifolds of the manifolds defining the bordism together with labels given by a set of defect conditions. Although the original bordism category had the structure of an ordinary category, equipping it with defect conditions naturally gives it the structure of a higher category.
Interesting examples of such field theories first appear in dimension two, and the authors go into some detail on the example of a lattice field theory. A nice application is given to centers of algebras. While the assignment of an algebra to its center is not in general functorial, the authors are able to give a functorial description in terms of bicategories.
The paper includes a concluding section on directions for further research.
For the entire collection see [Zbl 1230.81005].