Lecture Notes: Please be aware that these are rough notes that likely have errors. Any corrections are welcome.
Lecture 1 (courtesy of Kalle Kytötä)
Lecture 2 (courtesy of Kalle Kytötä)
Lecture 3 and 4 (Lectures and notes by Steven Flores)
Books: (please contact me if you have trouble locating any of these)
I. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, 1989
Yi-Zhi Huang, Two-Dimensional Conformal Geometry and Vertex Operator Algebras, 1997
J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, 2004
U. Tillmann (editor), Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal (London Mathematical Society Lecture Note Series). [Note that Graeme Segal's famous preprint on the definition of CFT from around 1987 is published in this volume with a new foreword/postscript]
E. Frenkel, Langlands Correspondence for Loop Groups (pre-publication version from his website)
M. Schottenloher, A Mathematical Introduction to Conformal Field Theory
E. Frenkel, D. Ben-Zvi, Vertex Algebras and Algebraic Curves: Second Edition.
Web resources:
Yi-Zhi Huang's blog on CFT. Good over view of the current status of the rigorous construction of CFT from VOA along with clarifications of long standing misunderstandings in the literature. "A number of people, including Rutgers graduate students, have asked me for a good introduction to two-dimensional conformal field theory. My answer to them is always: No, there are no good mathematical introductory books or expositions available. Even I myself am very unhappy with this answer". Huang is writing a book to address this problem. It will be very useful.
A. Weekes, Talk 3: Operator Product Expansions.
Articles:
Terry Gannon, Vertex Operator Algebras. In the Princeton Guide to Mathematics. (A nice short 11 page overview of VOA. Higly recommended)
P. Goddard, Meromorphic conformal field theory. In Infinite Dimensional Lie Algebras and Groups: Proceedings of the Conference. 1998
R. Griess and C. Lam. A new existence proof of the Monster by VOA theory, The Michigan Mathematical Journal, 2012. (An easier proof).
Yi-Zhi Huang, Riemann surfaces with boundaries and the theory of vertex operator algebras, in: Vertex Operator Algebras in Mathematics and Physics, ed. S. Berman, Y. Billig, Y.-Z. Huang and J. Lepowsky, Fields Institute Communications, Vol. 39, Amer. Math. Soc., Providence, 2003, 109--125.
Yi-Zhi Huang, "Geometric interpretation of vertex operator algebras", Proc. Nat. Acad. Sci., vol. 88, no. 22, 1991