Our article is a summary of some results for Riemannian manifolds that were obtained in \cite{gpes}-\cite{Pesssubm}. To the best of our knowledge these are the pioneering papers which contain the most general results about frames, Shannon sampling, and cubature formulas on compact and non-compact Riemannian manifolds. In particular, the paper \cite{gpes} gives an "end point" construction of tight localized frames on homogeneous compact manifolds. The paper \cite{Pessubm} is the first systematic development of localized frames on compact domains in Euclidean spaces.