  We compute production rates for two, three, four and five jets in
electron-positron annihilation at the third order in the QCD coupling constant.
At this order, three-jet production is described to next-to-next-to-leading
order (NNLO) in perturbation theory while the two-jet rate is obtained at
next-to-next-to-next-to-leading order (N$^3$LO). Our results yield an improved
perturbative description of the dependence of jet multiplicity on the jet
resolution parameter, $\ycut$, particularly at small values of $\ycut$.
