This is an expository talk centered on Brown-Douglas-Fillmore theory. We call an operator A on a separable Hilbert space essentially normal if AA*-A*A is a compact operator. It is easy to check that the sum of a normal and a compact is essentially normal. The opposite is not true — for example, unilateral right shift on l^2(N) is essentially normal, although not a sum of a normal and compact. I will state the criteria of an essentially normal operator being a sum of normal and compact, and show how extensions by compact operators come into play in a more general theory.