  Many, if not most, inflationary models predict the power-law index of the
spectrum of density perturbations is close to one, though not precisely equal
to one, |n-1| \sim O(0.1), implying that the spectrum of density perturbations
is nearly, but not exactly, scale invariant. Some models allow n to be
significantly less than one (n \sim 0.7); a spectral index significantly
greater than one is more difficult to achieve. We show that n \approx 1 is a
consequence of the slow-roll conditions for inflation and ``naturalness,'' and
thus is a generic prediction of inflation. We discuss what is required to
deviate significantly from scale invariance, and then show, by explicit
construction, the existence of smooth potentials that satisfy all the
conditions for successful inflation and give $n$ as large as 2.
