  This paper gives a survey of methods for the construction of space-frequency
concentrated frames on Riemannian manifolds with bounded curvature, and the
applications of these frames to the analysis of function spaces. In this
general context, the notion of frequency is defined using the spectrum of a
distinguished differential operator on the manifold, typically the
Laplace-Beltrami operator. Our exposition starts with the case of the real
line, which serves as motivation and blueprint for the material in the
subsequent sections.
  After the discussion of the real line, our presentation starts out in the
most abstract setting proving rather general sampling-type results for
appropriately defined Paley-Wiener vectors in Hilbert spaces. These results
allow a handy construction of Paley-Wiener frames in $L_2(\mfd{M})$, for a
Riemann manifold of bounded geometry, essentially by taking a partition of
unity in frequency domain. The discretization of the associated integral
kernels then gives rise to frames consisting of smooth functions in
$L_2(\mfd{M})$, with fast decay in space and frequency. These frames are used
to introduce new norms in corresponding Besov spaces on $\mfd{M}$.
  For compact Riemannian manifolds the theory extends to $L_p$ and Besov
spaces. Moreover, for compact homogeneous manifolds, one obtains the so-called
product property for eigenfunctions of certain operators and proves a cubature
formulae with positive coefficients which allow to construct Parseval frames
that characterize Besov spaces in terms of coefficient decay.
  Throughout the paper, the general theory is exemplified with the help of
various concrete and relevant examples, such as the unit sphere and the
Poincar\'{e} half plane.
