In 1937, Baer solved the classification problem for the torsion-free abelian groups of rank 1. Since then, despite the efforts of many mathematicians, no satisfactory solution has been found of the classification problem for the torsion-free abelian groups of rank n >1. So it is natural to ask whether the classification problem for the higher rank groups is genuinely more difficult. In this talk, I will explain how this question can be answered, using ideas from descriptive set theory and superrigidity theory.