The present paper deals with band-dominated operators on \(l^p(\mathbb Z)\), \(1<p<\infty \). It is proved that a band operator which is Fredholm on \(l^p(\mathbb Z)\) is also Fredholm on \(l^q(\mathbb Z)\) for all \(1<q< \infty\), and the index is independent of \(q\). Some generalizations for band-dominated operators are given. The index formula for band-dominated operators in terms of local indices of their limit operators is derived. This formula is applied to verify the stability of the finite section method for invertible band-dominated operators with slowly oscillating coefficients.