The paper surveys the categorical approach to universal algebra based on Lawvere’s concept of an algebraic theory. The corresponding concept in universal algebra is that of a clone. The paper stresses the role of reflexive coequalizers and arrives to Lawvere’s characterization of varieties of universal algebras (viewed as categories). There is also shown how to characterize quasivarieties of universal algebras and how to extend the theory to many-sorted universal algebras. The related concepts of regular, exact, or Mal’cev categories are included as well. Finally, there are considered localizations of varieties which are most known in the additive case, i.e., for varieties of modules over a ring.
For the entire collection see [Zbl 1034.18001].