Extended 3-dimensional topological field theories

I will survey recent advances in our understanding of extended 3-dimensional topological field theories. I will describe recent work (joint with B. Bartlett, C. Douglas, and J. Vicary) which gives an explicit ”generators and relations” classification of partially extended 3D TFTS (assigning values only to 3-manifolds, surfaces, and 1-manifolds). This will be compared to the fully-local case (which has been considered in joint work with C. Douglas and N. Snyder).