Norman Steenrod once said: "If you are having trouble explaining something, perhaps it is because you have not understood it well enough."  Recent developments have led to better understanding of  the proof of the best possible exponent for the homotopy groups of an odd primary Moore space.  Thus for the first time it has become realistic to lecture on the proof.  In so doing, it is necessary to give a quick outline of the work of Cohen, Moore, and Neisendorfer.