Ext(A,B) carries a natural group structure and one of the striking problems in abelian group theory is to determine completely the structure of Ext(A,B) for various groups A and B. In this book we will concentrate on the case of abelian groups and in some cases on modules over subrings of the rational numbers. In particular we ask, when Ext(A,B) is torsion-free. This book contains answers for finite rank groups A and B and also sheds some light on the infinite rank case, e.g. a generalization of coseparability. Furthermore we introduce the reader in the concept of torsion-free pairs, a generalization of Salce''s cotorsion-theories.