  The purpose of this paper is to establish some Adams-Moser-Trudinger
inequalities, which are the borderline cases of the Sobolev embedding, in the
hyperbolic space $\mathbb H^n$. First, we prove a sharp Adams inequality of
order two with the exact growth condition in $\mathbb H^n$. Then we use it to
derive a sharp Adams-type inequality and an Adachi-Tanaka-type inequality. We
also prove a sharp Adams-type inequality with Navier boundary condition on any
bounded domain of $\mathbb H^n$, which generalizes the result of Tarsi to the
setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an
improved Adams-type inequality in the spirit of Lions in $\mathbb H^n$. Our
proofs rely on the symmetrization method extended to hyperbolic spaces.
