This survey aims to accumulate all known results about semigroup rings, systematizing them around some central themes. It is planned to have 3 parts. The first part is devoted to the multiplicative semigroup of semigroup rings, and begins with a list of unified notations and terminology. After comments on three definitions of semigroup ring and its connections with the notion of contracted semigroup ring the author gives a detailed distribution of cited papers among the topics discussed in this survey. After these preliminary remarks many unsolved and solved problems are presented. It is impossible to list them all. Some are too general, but there are a lot of more concrete questions also. Among them are questions analogous to those for group rings: e.g. what semigroup rings have trivial units only, or what can be said about the supporting subsemigroup of the identity? In the semigroup case the answers may be quite different from those expected in analogy with the group case: e.g. the question ``for which semigroup rings is the supporting subsemigroup of any central idempotent finite?'' is nontrivial even for semilattices of groups. The results of Pot'jomkin and Rukolajne on the centers of semigroup rings are presented in detail. The deepest results mentioned in the survey are those of L. Bokut' (1964-69): the construction of a semigroup ring without zero-divisors not embedabble into a skew field, and of a semigroup ring not embeddable into an sfield though its multiplicative semigroup is embeddable into a group. The paper ends with a very extensive (253 items) and useful list of references. The second and third parts of this survey are designed to cover primitivity, radicals and semisimplicity of semigroup rings, linear representations and modules over semigroup rings, characters and various other results about semigroup rings.