The study of Lefschetz properties of artinian algebras has its origin in the Hard Lefschetz Theorem and was encouraged by Stanley’s use of it in the proof of the g-Conjecture for simplicial polytopes. The Weak Lefschetz Property means that multiplication by a general linear form has maximal rank in each degree and the Strong Lefschetz Property that the same also holds for all powers of the form. Despite the fact that most algebras have these property it is surprisingly hard to prove general results in this direction. I will discuss some of the known results and some of the open problems in this area.