Abstract: I will start by reviewing my recent work with Dan Grady on the geometric cobordism hypothesis and locality of fully extended nontopological functorial field theories. I will then apply these results to explicitly compute, in terms of homotopy coherent representations of Lie groups, the space of 2-dimensional fully extended functorial field theories with geometric structures like flat 2-dimensional Riemannian metrics, 2|1-dimensional super Euclidean structures, or conformal structures. If time permits, I will discuss further constructions of field theories (ongoing work with Dan Grady) that involve differential characteristic classes, index and eta-invariants, and quantization.