Let \(E/F\) be a finite Galois extension of fields. An element \(w\in E\) is called normal in \(E/F\) if \(w\) and its Galois conjugates form a basis for \(E\) as an \(F\)-vector space. Let \(\mathbb Q_n\) denote the \(n\)th cyclotomic field. An element \(w\in \mathbb Q_e\) is called universally normal in the extension \(\mathbb Q_e/\mathbb Q_d\) if \(w\) is normal for every intermediate cyclotomic extension \(\mathbb Q_e/\mathbb Q_n\), and it is called completely normal if \(w\) is normal for every intermediate extension \(\mathbb Q_e/K\). The author characterizes and explicitly constructs completely normal elements for \(\mathbb Q_{r^m}/\mathbb Q\), where \(r\) is an odd prime and where \(m\geq 1\). Universally normal elements are constructed and characterized for \(\mathbb Q_{2^m}/\mathbb Q\) for \(m\geq 2\). Finally, the author studies trace-compatible sequences of universally normal elements, namely sequences of elements \(w_n\in \mathbb Q_n\), with \(n\) varying, that are universally normal for \(\mathbb Q_n/\mathbb Q\) and such that, when \(d\) divides \(e\), the trace from \(\mathbb Q_e\) to \(\mathbb Q_d\) of \(w_e\) is \(w_d\).