  In this paper we extend the bump conjecture and a particular case of the
separated bump conjecture with logarithmic bumps to iterated commutators
$T_b^m$. Our results are new even for the first order commutator $T_b^1$. A new
bump type necessary condition for the two-weighted boundedness of $T_b^m$ is
obtained as well. We also provide some results related to a converse to Bloom's
theorem.
