  Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$
strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n'
= [WB_n, WB_n]$ respectively. We obtain a set of generators and defining
relations for these commutator subgroups. In particular, we prove that $VB_n'$
is finitely generated if and only if $n \geq 4$, and $WB_n'$ is finitely
generated for $n \geq 3$. Also we prove that $VB_3'/VB_3'' =\mathbb{Z}_3 \oplus
\mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z}^{\infty}$, $VB_4' / VB_4'' =
\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$, $WB_3'/WB_3'' =
\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z},$
$WB_4'/WB_4'' = \mathbb{Z}_3,$ and for $n \geq 5$ the commutator subgroups
$VB_n'$ and $WB_n'$ are perfect, i.e. the commutator subgroup is equal to the
second commutator subgroup.
