  Isentropic and isocurvature axion inflationary perturbations are studied in
detail. If isocurvature perturbations are present, COBE requires the
inflationary Hubble constant to be rather small, $H_{I} \stackrel{<}{\sim}
10^{12}$GeV, regardless of the inflation model. Microwave background
anisotropies are calculated for CDM models with a mixture of axion and
supersymmetric CDM. The angular power spectrum of the microwave background
depends not only on the amplitude of the isocurvature power spectrum but also
on $\Omega_{a}$. The Saskatoon data and $\sigma_{8}$ are used to constrain the
contribution of axion isocurvature perturbations. Consistency with the
Saskatoon data requires the square of the amplitude of the isocurvature
perturbations to be $\stackrel{<}{\sim} 0.1$ times that of the adiabatic
perturbations at $\ell=87$, and $\stackrel{<}{\sim} 10^{-3}-10^{-2}$ that of
the adiabatic perturbations at $\ell=237$. Although not dominant, isocurvature
perturbations are not yet ruled out. The consequences of isocurvature
perturbations for axions and inflation are discussed.
