The $(\infty,n)$-category of cobordisms

I will explain how to construct the symmetric monoidal $(\infty,n)$-category of
cobordisms using (complete) $n$-fold Segal spaces as a model, following work by
Galatius-Madsen-Tillmann-Weiss and Lurie. Its homotopy category and bicategory
recover the usual cobordism category and the cobordism bicategory defined by
Schommer-Pries. Finally, this allows to give a precise definition and explicitly
construct examples of fully extended topological field theories in the sense of
Lurie's formulation of the Cobordism Hypothesis.