The conditions unlder which a semnigroup ring is regu- lar (in the sense of von Neumann) are inlvestigated. SLufficient con- ditions are obtained in order that the semnigroup ring of an inverse semigroup be regular. Consequences of the regularity of the seni- group ring for the subgroups of the semigroup are established. The two results are then used to find necessary and sufficien-t conditions for the regularity of the semigroup ring when the semigroup is inverse and a union of groups. 1. Introduction. Regularity of group rings has been investigated by Auslander (1), Connell (31 and McLaughlin (4). In this paper we attempt to extend their results to semigroup rings. Throughout this paper R will denote an associative ring having an identity element. If R is a ring and D is a semigroup RD will denote the semigroup ring of D over R and (RD)o the contracted semiigroup ring if D has a zero. The definition of (contracted) seniigroup rings is given in (8, ?3). It is analogous to the definition of semnigroup algebra