The Brown-Douglas-Fillmore Theorem is originally for the classification of essential normal operators which later for other special essential extensions. It was later developed in greater generality for all essential extensions by Kasparov and to the KK-theory. Kasparov's KK-theory is for stable C*-algebras and for stably unitary equivalences. Recent development in the Elliott program for simple leads to questions such as how to classify C*-algebras with single essential ideals (which may not be stable).

In this talk, we would like to have an elementary introduction to the question why one needs to revisit extension theory after the development of KK-theory. We plan to present some more recent results with Ping Ng and with Jamie Gabe and Ping.