This thesis introduces a higher-order optimization method for solving non-smooth variational problems on Riemannian manifolds. In this work, we apply the Riemannian Semismooth Newton (RSSN) method to a non-smooth non-linear optimality system derived in recent advances in manifold duality theory. In particular we will show a novel local convergence result for an inexact version of the Riemannian Semismooth Newton method and show state-of-the-art performance in numerical experiments by solving several `2-TV-like problems on manifolds with positive and negative curvature.