This paper deals with questions related to the nil radical and the Jacobson radical of the endomorphism rings of torsion-free Abelian groups. The most complete results are obtained for groups of finite rank. A characterization is given for the Jacobson radical of the endomorphism ring of a torsion-free Abelian group of finite rank. The question of when the Jacobson radical of the endomorphism ring of a torsion-free Abelian group of finite rank is nilpotent (equal to zero) is completely settled.Bibliography: 7 titles.