When the "left-overs" are what counts

Abstract: Everyone who cooks knows that sometimes the left-overs are the best part of the meal. There is an analogue in algebraic geometry called "residual intersection theory", in which some interesting object is represented as what is "left over" after we subtract something known from a known intersection.  Historically, this problem arose in at least three contexts:

 *) The question of how many conics are tangent to 5 given general conics in the plane?
 *) Brill and Noether's analysis of the Riemann-Roch theorem
 *) Halphen's analysis of space curves

 Even understanding what is meant by "left over" in these contexts has lead to the invention of some deep and important mathematics. I'll explain these three sources and their modern extensions, and talk about open problems at the frontier of the theory.