We study stochastic flow processes and diffusions on smooth Riemannian manifolds starting from the well-known notion of a flow to a vector field. As an application we sketch how these concepts can be used to give a very simple proof for existence and uniqueness of a solution to the Dirichlet problem. We give a brief overview how to define Brownian motion on (smooth Riemannian) manifolds: the extrinsic approach as solution to the usual martingale problem using a Whitney embedding and the Eells-Elworthy-Malliavin approach using the projection from the orthonormal frame bundle. All notions will be briefly introduced during the talk as needed concerning the broad audience.