Let G be a finite group and let L/K be a Galois G-extension. The field L is called K-adequate if there is an element in the relative Brauer group Br(L/K) of index [L:K]. In such case there is a K-central division algebra D for which L is a maximal subfield of D. Thus D is a crossed product, D=(L/K,f), and G is called K-admissible. We shall discuss the problem of determining which groups are K-admissible over a number field K, starting with tamely ramified extensions which are K-adequate and proceeding to a discussion of wild admissibility. This problem has necessary realizability conditions (over the number field and over some of its completions). We shall see that these conditions are (frequently but) not always sufficient. We shall apply the acquired knowledge to discuss the following equivalence problem: Given two number fields K_1,K_2 that have the same admissible groups, are K_1 and K_2 necessarily isomorphic?