Summary: It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups \(C_{p_i}, p_i \in \mathbb{N}\), is equal to 1. The commutator width of direct limit of wreath product of cyclic groups is found. This paper gives upper bounds of the commutator width \((cw(G))\) [A. Muranov, Int. J. Algebra Comput. 17, No. 3, 607–659 (2007; Zbl 1141.20022)] of a wreath product of groups. A presentation in the form of wreath recursion [V. Nekrashevych, Self-similar groups. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1087.20032)] of Sylow \(2\)-subgroups \(\mathrm{Syl}_2 A_{2^k}\) of \(A_{2^k}\) is introduced. As a corollary, we obtain a short proof of the result that the commutator width is equal to 1 for Sylow 2-subgroups of the alternating group \(A_{2^k}\), where \(k > 2\), permutation group \(S_{2^k}\) and for Sylow \(p\)-subgroups \(\mathrm{Syl}_2 A_{p^k}\) and \(\mathrm{Syl}_2 S_{p^k}\). The commutator width of permutational wreath product \(B \wr C_n\) is investigated. An upper bound of the commutator width of permutational wreath product \(B \wr C_n\) for an arbitrary group \(B\) is found.