Since R. Baer introduced in 1933 the functor Ext in abelian group theory, it has been considered extensively in the literature (see e.g. the books by Fuchs and Eklof-Mekler). Recall that the Ext-functor is the first derived functor of the Hom-functor but Ext(A,B) can also be thought of as the set of all equivalence classes of short exact sequences of the form 0 -> B -> C -> A -> 0, thus classifies all extensions of the group B by the group A. Ext(A,B) carries a natural group structure and one of the striking problems in abelian group theory and also in this thesis is to determine completely the structure of Ext(A,B) for various groups A and B. In particular, the question when Ext vanishes has achieved much attention. In this context the famous Whitehead-problem asks whether every abelian group $G$ satisfying Ext(G,\Z)=0 has to be free. For countable groups this is true by a result due to Stein in 1951 and independently to Ehrenfeucht in 1955, but the general result had been open for many decades until Saharon Shelah proved its independence in ZFC in 1977. Going one step further, the more general and (as we will see) much more complicated question is: When is Ext(A,B) torsion-free for abelian groups A and B?