The present work clarifies the relation between domains of universal machines and r.e. prefix-free supersets of such sets. One such characterisation can be obtained in terms of the spectrum function s _W (n) mapping n to the number of all strings of length n in the set W. An r.e. prefix-free set W is the superset of the domain of a universal machine iff there are two constants c,d such that s _W (n) + s _W (n + 1) + ... + s _W (n + c) is between 2 ^{n − H(n) − d} and 2 ^{n − H(n) + d} for all n; W is the domain of a universal machine iff there is a constant c such that for each n the pair of n and s _W (n) + s _W (n + 1) + ... + s _W (n + c) has at least the prefix-free Description complexity n. There exists a prefix-free r.e. superset W of a domain of a universal machine which is the not a domain of a universal machine; still, the halting probability Ω _W of such a set W is Martin-Löf random. Furthermore, it is investigated to which extend this results can be transferred to plain universal machines.