This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere S^d (directional spaces), real projective space RP^{N-1} (axial spaces), complex projective space CP^{k-2} (planar shape spaces) w.r.t. Veronese-Whitney embeddings and a three-dimensional shape space Σ_3^4.