  We define a scale of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$,
$p\in[1,\infty]$, that are invariant under suitable Fourier integral operators
of order zero. This builds on work by Smith for $p=1$. We also introduce a
notion of off-singularity decay for kernels on the cosphere bundle of
$\mathbb{R}^{n}$, and we combine this with wave packet transforms and tent
spaces over the cosphere bundle to develop a full Hardy space theory for
oscillatory integral operators. In the process we extend the known results
about $L^{p}$-boundedness of Fourier integral operators, from local boundedness
to global boundedness for a larger class of symbols.
