  We investigate the palindromic width of finitely generated solvable groups.
We prove that every finitely generated $3$-step solvable group has finite
palindromic width. More generally, we show the finiteness of palindromic width
for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary
solvable groups of step $\geq 3$, we prove that if $G$ is a finitely generated
solvable group that is an extension of an abelian group by a group satisfying
the maximal condition for normal subgroups, then the palindromic width of $G$
is finite. We also prove that the palindromic width of $\mathbb Z \wr \mathbb
Z$ with respect to the set of standard generators is $3$.
