  We introduce the concept of Calder\'on-Zygmund inequalities on Riemannian
manifolds. For $1<p<\infty$, these are inequalities of the form $$ \left\Vert
\mathrm{Hess}\left( u\right) \right\Vert _{L^p}\leq C_{1}\left\Vert
u\right\Vert _{L^p}+C_{2}\left\Vert \Delta u\right\Vert _{L^p}, $$ valid a
priori for all smooth functions $u$ with compact support, and constants
$C_1\geq 0$, $C_2>0$. Such an inequality can hold or fail, depending on the
underlying Riemannian geometry. After establishing some generally valid facts
and consequences of the Calder\'on-Zygmund inequality (like new denseness
results for second order $\mathsf{L}^p$-Sobolev spaces and gradient estimates),
we establish sufficient geometric criteria for the validity of these
inequalities on possibly noncompact Riemannian manifolds. These results in
particular apply to many noncompact hypersurfaces of constant mean curvature.
