ABSTRACT: 

        In the early 1930's, Garrett Birkhoff introduced the
notion a variety  V,  i.e. the class of all algebras (in the general
sense) that model a fixed set  E  of equations, and proved his famous
theorem characterizing such equational classes being classes closed
under the formation of subalgebras, homomorphic images and products.
Ever since then (and more intensely since 1970), such classes have been
actively studied --- both in general and in many particular examples.

        In this talk we will survey some of this work, mentioning important
results and problems of Tarski, Jonsson, McKenzie, Baker. We will also
include some recent material on the modeling of the equations  E  by
continuous operations on a topological space  A. (Birkhoff was interested
in this question as well.)  Some recent results of the author say that
many simple spaces  A,  e.g. a surface of genus 2, cannot continously
model any except the most trivial of equations  E.