We extend Yamagami and Matsui's theory of the Kaprekar transformation in base 3 to higher odd bases. The structure of primitive proper, primitive non-proper and general cycles appears in all odd bases, but many other features differ according to whether the base is congruent to 1 or 3 modulo 4. In the latter case, cycles are derived from subgroups and cosets in multiplicative groups modulo odd numbers. We examine cycles and fixed points in bases 5 and 7 in some detail, and make some broad observations relating to higher bases. There also exist many cycles outside the primitive/general structure.