Title:  Everything I knew (and maybe that you know) about the Stieltjes integral is wrong.

Abstract:  I will discuss three definitions of the Riemann integral (the Riemann integral, the Darboux integral, and a variant of the Darboux integral) and why they are equivalent.  I will then introduce the generalizations of these results to define the Stieljtes integral and explain where and how they diverge.  I will touch briefly on some lost history of the Stieltjes integral, and show how these definitions evolved.   In particular, I will give a characterization of Stieltjes integrability that captures (I think) everything I have learned.