In these three lectures, my goal is to give a very fast and elementary introduction to some basic aspects of quantum field theory and how the A, B model topological field theories are built by physicists as special quantum field theories. In the first lecture, I'll give a quick introduction to quantum field theory, outlining where Feynman diagrams come from, and the idea of the renormalization group. In the second and third lectures, I'll specialize to nonlinear sigma models, talk about their definition, perturbation theory, and supersymmetric versions thereof, and then outline the A, B model topological field theories as twisted supersymmetric nonlinear sigma models. If time remains, I may briefly outline the open string B model and how derived categories arise in it.