An algebra is said to be abelian if it satisfies all sentences of the form

where t is a term. In our talk we discuss recent results about the structure of abelian algebras and of locally finite varieties where all members are abelian.

In the first part of our talk we will discuss the relationship between abelian, affine and quasi-affine algebras. We characterize affine algebras as precisely those abelian algebras which generate a variety satisfying an idempotent Mal'cev condition which fails to hold in the variety of semilattices.

In the second part of the talk we discuss the local structure of algebras in locally finite abelian varieties, and we explain why such varieties must be finitely generated. We give an example showing that locally finite abelian varieties may be nonfinitely based, but we go on to show that they cannot be inherently nonfinitely based.