We discuss some known and introduce some new hierarchies andreducibilities on regular languages, with the emphasis on thequantifier-alternation and difference hierarchies of thequasi-aperiodic languages. The non-collapse of these hierarchies anddecidability of some levels are established. Complete sets in thelevels of the hierarchies under the polylogtime and somequantifier-free reducibilities are found. Some facts about thecorresponding degree structures are established. As an application,we characterize the regular languages whose balanced leaf-languageclasses are contained in the polynomial hierarchy. For anydiscussed reducibility we try to give motivations and openquestions, in a hope to convince the reader that the study of thesereducibilities is interesting for automata theory and computationalcomplexity.