CFT/QFT/TFT/etc. is a huge subject...
Here are some random references off the top of my head...
Segal, "The definition of conformal field theory".
Costello, "Topological conformal field theories and Calabi-Yau categories" -- This is (essentially) the 2d version of the (Hopkins-)Lurie/Baez-Dolan cobordism hypothesis that Lennart mentions. See also Kontsevich-Soibelman, "Notes on A-infinity...". This stuff is closely related to mirror symmetry, which is - in physics terms - a duality between certain field theories (or sigma models). Mirror symmetry by itself is already a huge enterprise...
See papers by Yi-Zhi Huang for stuff about vertex operator algebras and CFTs.
One can consider string topology from a field theory viewpoint... see for example Sullivan, "String Topology: Background and Present State" and Blumberg-Cohen-Teleman, "Open-closed field theories, string topology, and Hochschild homology". This is actually related to the work of Costello, Lurie, Kontsevich mentioned above -- see e.g. section 2.1 of Costello's paper.
An important problem is that of making rigorous some of the things that physicists do in QFT, such as path integrals. See Costello, "Renormalization and effective field theory" and also Borcherds, "Renormalization and quantum field theory".
There's also Chern-Simons theory... Gromov-Witten theory... Kapustin-Witten theory... Rozansky-Witten theory...
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