  In this lecture we explain the intimate relationship between modular
invariants in conformal field theory and braided subfactors in operator
algebras. Our analysis is based on an approach to modular invariants using
braided sector induction ("$\alpha$-induction") arising from the treatment of
conformal field theory in the Doplicher-Haag-Roberts framework. Many properties
of modular invariants which have so far been noticed empirically and considered
mysterious can be rigorously derived in a very general setting in the subfactor
context. For example, the connection between modular invariants and graphs (cf.
the A-D-E classification for $SU(2)_k$) finds a natural explanation and
interpretation. We try to give an overview on the current state of affairs
concerning the expected equivalence between the classifications of braided
subfactors and modular invariant two-dimensional conformal field theories.
