  We investigate models of `intermediate' inflation, where the scale factor
$a(t)$ grows as $a(t) = \exp (A t^f)$, $0 < f < 1$, $A$ constant. These
solutions arise as exact analytic solutions for a given class of potentials for
the inflaton $\phi$. For a simpler class of potentials falling off as a power
of $\phi$ they arise as slow-roll solutions, and in particular they include,
for $f = 2/3$, the class of potentials which give the Harrison--Zel'dovich
spectrum. The perturbation spectral index $n$ can be greater than unity on
astrophysical scales. It is also possible to generate substantial gravitational
waves while keeping the scalar spectrum close to scale-invariance; this latter
possibility performs well when confronted with most observational data.
