  We apply the theory of de Branges-Rovnyak spaces to describe kernels of some
Toeplitz operators on the classical Hardy space $H^2$. In particular, we
discuss the kernels of the operators $T_{\bar f/ f}$ and $T_{\bar I\bar f/ f}$,
where $f$ is an outer function in $H^2$ and $I$ is inner such that $I(0)=0$. We
also obtain results on de Branges-Rovnyak spaces generated by nonextreme
functions.
