We investigate models of `intermediate' inflation, where the scale factor $a\left(t\right)\mathrm{}$ grows as $a\left(t\right)=\exp \left(A{t}^f\right)$, $0<f<\mathrm{}1\mathrm{}$, $A$ 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 $f=\frac{2}{3}$, 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.

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