The Reidemeister number of an automorphism ϕ of an Abelian group G is calculated by determining the cardinality of the quotient group G/(ϕ − 1G)(G), and the Reidemeister spectrum of G is precisely the set of Reidemeister numbers of the automorphisms of G. In this work we determine the full spectrum of several types of group, paying particular attention to groups of torsion-free rank 1 and to direct sums and products. We show how to make use of strong realization results for Abelian groups to exhibit many groups where the Reidemeister number is infinite for all automorphisms; such groups then possess the so-called R∞-property.We also answer a query of Dekimpe and Gonçalves by exhibiting an Abelian 2-group which has the R∞-property
