Ideals and their Fitting ideals

For an ideal $I$ in a Noetherian ring $R$, the Fitting ideals ${\text{Fitt}}_j(I)$ are studied. We discuss the question of when ${\text{Fitt}}_j(I)=I$ or $\sqrt{{\text{Fitt}}_j(I)}=\sqrt{I}$ 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.