Consider a permutation $\tau$ of the set $\{1,2,\dots,n,\}$.  If we divide the unit interval $[0,1)$ into $n$ half-open subintervals, we can consider the map $f$ which rearranges the subinterval according to the permutation $\tau$.  Such maps are called interval exchange transformations (IETs) and are the order preserving piecewise isometries of intervals, and preserve the Lebesgue measure.  IETs were first studied by Sinai in 1973, and then Keane in 1977, who showed that each minimal IET had a finite number of ergodic measures and conjectured that the Lebesgue measure was in fact the only ergodic invariant measure for such maps.  Much of the following research on IETs was based around proofs of this conjecture and will be discussed in the talk.