  We show that the wreath product $G \wr \mathbb{Z}^n$ of any finitely
generated group $G$ with $\mathbb{Z}^n$ has finite palindromic width. We also
show that $C \wr A$ has finite palindromic width if $C$ has finite commutator
width and $A$ is a finitely generated infinite abelian group. Further we prove
that if $H$ is a non-abelian group with finite palindromic width and $G$ any
finitely generated group, then every element of the subgroup $G' \wr H$ can be
expressed as a product of uniformly boundedly many palindromes. From this we
obtain that $P \wr H$ has finite palindromic width if $P$ is a perfect group
and further that $G \wr F$ has finite palindromic width for any finite,
non-abelian group $F$.
