Acts over semigroups

We present a review of results obtained mainly in the last two decades in a number of areas of the theory of acts over semigroups. The authors limited themselves to the structural theory of acts. The acts over completely ($0$-)simple semigroups, with conditions on the congruence lattice, diagonal acts, biacts and multiacts, and also partial acts are considered. Our work is an expanded version of the report made by the authors in October 2017 at the conference of Institute of Mathematics of Technical University of Berlin, dedicated to the 75th anniversary of Professor Ulrich Knauer, supplemented by the results of later works.