  We give an exposition on the current status of classification of operator
algebraic conformal field theories. We explain roles of complete rationality
and alpha-induction for nets of subfactors in such a classification and present
the current classification result, a joint work with R. Longo, for the case
with central charge less than 1, where we have a complete classification list
consisting of the Virasoro nets, their simple current extensions of index 2,
and four exceptionals. Two of the four exceptionals appear to be new.
