As is well known, a complex $m\times m$ matrix $A$ is a commutator (i.e., there are matrices $B$ and $C$ of the same dimensions as $A$ such that $A=[B,C]=BC-CB$) if and only if $A$ has zero trace. If $\|\cdot\|$ is the operator norm from $\ell_2^m$ to itself and $|\cdot|$ any ideal norm on $m\times m$ matrices then clearly for any $A,B,C$ as above $|A|\le 2\|B\||C|$. Does the converse hold? That is, if $A$ has zero trace are there $m\times m$ matrices $B$ and $C$ such that $A=[B,C]$ and $\|B\||C|\le K|A|$ for some absolute constant $K$? If not, what is the behavior of the best $K$ as a function of $m$? The talk will concentrate on two recent results on this problem. The first is a couple of years old result of Johnson, Ozawa and myself which gives some partial answers to this problem for the most interesting case of $|\cdot|=\|\cdot\|$. The second is a more recent result of Angel and myself which solves the problem for $|\cdot|=$ the Hilbert--Schmidt norm.