This project began at the kind suggestion of Professor R. Remmert that I consider writing an Ergebnisse Bericht on the general theme of harmonic analysis on compact homogeneous spaces. For a number of reasons, I felt that a less formal treatment of the material would be more appropriate. Perhaps the crucial thing for me was that I did not feel that there was a sufficient body of generally interesting results in essentially final form to justify an Ergebnisse monograph.