3-dimensional defect TQFTs and their tricategories

I will discuss 2- and 3-dimensional "defect TQFTs", which are symmetric monoidal functors on stratified and decorated bordism categories. Such functors give rise to "pivotal 2-categories" and "tricategories with duals", respectively. After a review of the relevant higher-categorical notions and their graphical calculus, I shall explain how they capture the algebraic essence of defect TQFTs. Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and they can be nontrivially extended to defect TQFTs with embedded surfaces. This talk is mostly based on joint work with Catherine Meusburger and Gregor Schaumann (arXiv:1603.01171).