  This survey describes some recent rigidity results obtained by the authors
for the prescribed mean curvature problem on graphs $u : M \rightarrow
\mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as
on graphical solitons for the mean curvature flow, in warped product ambient
spaces. A detailed analysis of the mean curvature operator is given, focusing
on maximum principles at infinity, Liouville properties, gradient estimates.
Among the geometric applications, we mention the Bernstein theorem for positive
entire minimal graphs on manifolds with non-negative Ricci curvature, and a
splitting theorem for capillary graphs over an unbounded domain $\Omega \subset
M$, namely, for CMC graphs satisfying an overdetermined boundary condition.
