http://nlab.mathforge.org/nlab/show/conformal+field+theory
http://mathoverflow.net/questions/37063/what-do-mathematicians-currently-do-in-conformal-field-theory-or-more-general-fi
http://mathoverflow.net/questions/77635/what-exactly-is-the-relation-between-string-theory-and-conformal-field-theory
For conformal field theory and other types of field theory: Check out the MO answers of Ben-Zvi.
nLab page on Conformal field theory