Given a Riemannian manifold (M,g), it was realized a long time ago that special maps from (M,g) to some metric spaces give important insights on the geometry of (M,g). Here “special” means isometric, harmonic or area minimizing etc. I will give a survey of such special mappings and some applications to systolic geometry, inverse problems, dynamical systems. I will also discuss new applications to stability problems for geometric inequalities, and to the construction of canonical shapes for manifolds.