AbstractGiven a cardinal$\lambda $with$\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality$\lambda $whose automorphism group is a free group of rank$2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank$2^\lambda $is the automorphism group of a field of cardinality$\lambda $whenever$\lambda $is a cardinal with$\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that$\lambda =\lambda ^{\aleph _0}$is not necessary for the existence of a field of cardinality$\lambda $whose automorphism group is a free group of rank$2^\lambda $. Finally, we use them to prove that the existence of a cardinal$\lambda $of uncountable cofinality with the property that there is no field of cardinality$\lambda $whose automorphism group is a free group of rank greater than$\lambda $implies the existence of large cardinals in certain inner models of set theory.