The Lefschetz properties

We introduce the Lefschetz properties of an endomorphism of a graded vector space. We intend to show that this has many applications to various areas of algebra, geometry and combinatorics, particularly to Artinian Gorenstein rings and cohomology rings of topological spaces. We give an overview on classical theorems, recent developments and open problems. Most results we discuss were obtained in the years 1980–2020, but some of them trace back as early as 1850s in a disguised form.