Some open problems in the structure theory of regular semigroups

The aim of this note is to summarise the most important results which generalise those describing the structure of inverse semigroups via semidirect products of semilattices by groups, and to formulate several open problems whose investigation might contribute to the development of this topic. This is the written version of the author's talk given at the problem session of the Colloquium on Semigroups, Szeged, 2000. The paper also contains some problems raised there by D. B. McAlister and P. A. Grillet. December 7, 2000