Abstract: An equivalence relation E on a standard Borel space is said to be {\it smooth} if the quotient object X/E can itself be thought of as a subset of a standard Borel space.

This amounts to saying that there is a Borel function assigning points in some standard Borel space as complete invariants for E, and as such can be thought of as a measure of classifiability. (This idea seems to have been first put forward in the study of infinite dimensional group representations -- with the suggestion that we should only consider the representations to be reasonably classifiable if the conjugacy equivalence relation is smooth.)

I will talk about the foundational aspects of this idea, the Harrington-Kechris-Louveau dichotomy theorem, and some recent work by set theorists who are trying to understand more general notions of classifiability.