A complex manifold is a contact manifold if there is a distribution in the tangent bundle which is as non-integrable as possible. I will report on recent progress in classification of projective contact manifolds focusing on the Fano case in (complex) dimension 7 and 9. Our work implies the classification of quaternion-Kaehler manifolds of (real) dimensions 12 and 16, a famous problem from Riemannian geometry. The tools we use include representation theory and actions of (complex) reductive groups on manifolds, symplectic geometry, characteristic classes, and equivariant localisation theorems.