In this talk we study the Hermite interpolation and approximation problem. It aims at producing a function together with its derivatives, which interpolate or approximate given discrete point-vector data. The classical Hermite method interpolates data in linear spaces using polynomial functions.  We are interested in interpolating or approximating manifold-valued point-vector data using curves defined solely by the intrinsic geometry of the underlying manifold. For this purpose we use iterative refinement algorithms, called Hermite subdivision schemes. These algorithms successively refine given point-vector data and, via a limit process, produce a function and its derivatives solving the Hermite problem. Subdivision algorithms are well-suited for our purpose, as they can be adapted easily from the linear situation to the manifold setting. We give an introduction to linear Hermite subdivision schemes and present an adaptation to manifolds using geodesics and parallel transport. Furthermore, we show that the resulting nonlinear algorithms solve the manifold-valued Hermite problem by providing convergence and smoothness analysis.