Applications of the Magnus embedding in the theory of varieties of groups and Lie algebras

We give a survey of results of the theory of varieties of groups and Lie algebras which were proved using the Magnus embedding or its generalizations (the Magnus embedding is the embedding of a group of the form $F/R'$ into the wreath product $A\operatorname{wr}F/R$, where $A$ is a free Abelian group). We exhibit short proofs of the embedding theorem as well as of the criterion for an element of the wreath product to belong to the embedded group.