  For an ideal $I$ in a Noetherian ring $R$, the Fitting ideals
$\textrm{Fitt}_j(I)$ are studied. We discuss the question of when
$\textrm{Fitt}_j(I)=I$ or $\sqrt{\textrm{Fitt}_j(I)}=\sqrt{I}$ for some $j$. A
classical case is the Hilbert-Burch theorem when $j=1$ and $I$ is a perfect
ideal of grade $2$ in a local ring.
