On a complete noncompact Riemannian manifold the existence of certain nontrivial harmonic functions can be used to discover important geometric properties of the manifold. One possible approach to monotonicity is to take the Bochner formula, which involves Ricci curvature, and integrate it on the sublevel sets of the harmonic function. This idea can be used to establish geometric inequalities (e.g., of Minkowski type for hypersurfaces) or volume estimates (e.g., of Bishop-Gromov type), among many others. We will survey the method and present some recent applications.