  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.
