  We study the stability of band preserving operators on Banach lattices. To
this end the notion of $\varepsilon$-band preserving mapping is introduced. It
is shown that, under quite general assumptions, a $\varepsilon$-band preserving
operator is in fact a small perturbation of a band preserving one. However, a
counterexample can be produced in some circumstances. Some results on automatic
continuity of $\varepsilon$-band preserving maps are also obtained.
