The achromatic index of the complete graph with n vertices is the maximum number of colors used to color its edges such that no adjacent edges are of the same color, and, for any two colors \(c_ 1\) and \(c_ 2\), there are two adjacent edges, one colored by \(c_ 1\) and the other colored by \(c_ 2\). Bounds, as well as precise values for small n, of the achromatic index are studied.