Suppose that T is an equational theory of groups or of rings. If T is finitely axiomatizable, then there is a least number μ so that T can be axiomatized by μ equations. This μ can depend on the operation symbols that occur in T. In the 1960s, Tarski and Green completely determined the values of μ for arbitrary equational theories of groups and of rings. While Tarski and Green announced the results of their collaboration in 1970, the only fuller (...) publication of their work occurred as part of a seminar led by Tarski at Berkeley during the 1968–69 academic year. The present paper gives a full account of their findings and their proofs. (shrink)