  A class of graphs closed under taking induced subgraphs is $\chi$-bounded if
there exists a function $f$ such that for all graphs $G$ in the class, $\chi(G)
\leq f(\omega(G))$. We consider the following question initially studied in [A.
Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em
Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a
$\chi$-bounded class $\cal C$, is the class $\bar{C}$ $\chi$-bounded (where
$\bar{\cal C}$ is the class of graphs formed by the complements of graphs from
$\cal C$)? We show that if $\cal C$ is $\chi$-bounded by the constant function
$f(x)=3$, then $\bar{\cal C}$ is $\chi$-bounded by
$g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for
every constant $c>0$, if $\cal C$ is $\chi$-bounded by a function $f$ such that
$f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $\chi$-bounded. For every $j$,
we construct a class of graphs $\chi$-bounded by $f(x)=x+x/\log^j(x)$ whose
complement is not $\chi$-bounded.
