  We establish sharp Hardy-Adams inequalities on hyperbolic space
$\mathbb{B}^{4}$ of dimension four. Namely, we will show that for any
$\alpha>0$ there exists a constant $C_{\alpha}>0$ such that \[
\int_{\mathbb{B}^{4}}(e^{32\pi^{2} u^{2}}-1-32\pi^{2}
u^{2})dV=16\int_{\mathbb{B}^{4}}\frac{e^{32\pi^{2} u^{2}}-1-32\pi^{2}
u^{2}}{(1-|x|^{2})^{4}}dx\leq C_{\alpha}. \] for any $u\in
C^{\infty}_{0}(\mathbb{B}^{4})$ with \[
\int_{\mathbb{B}^{4}}\left(-\Delta_{\mathbb{H}}-\frac{9}{4}\right)(-\Delta_{\mathbb{H}}+\alpha)u\cdot
udV\leq1. \]
  As applications, we obtain a sharpened Adams inequality on hyperbolic space
$\mathbb{B}^{4}$ and an inequality which improves the classical Adams'
inequality and the Hardy inequality simultaneously. The later inequality is in
the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two
given by Wang and Ye [37] and on any convex planar domain by the authors [26].
  The tools of fractional Laplacian, Fourier transform and the Plancherel
formula on hyperbolic spaces and symmetric spaces play an important role in our
work.
