This monograph has arisen in part from E. Witten's lectures on topological quantum field theory given in the spring of 1989 at Princeton University. At that time, Witten unified several important mathematical works in terms of quantum field theory, most notably the Donaldson polynomial, the Gromov-Floer homology and the Jones polynomials. In this book, Sen Hu has added material to provide some of the details left out of Witten's lectures and to update some new developments. In Chapter Four he presents a construction of knot invariant via representation of mapping class groups based on the work of Moore-Seiberg and Kohno. In Chapter Six he offers an approach to constructing knot invariant from string theory and topological sigma models proposed by Witten and Vafa. In addition, relevant material by S.S. Chern and E. Witten has been included as appendices for the convenience of readers.