  A continuous frame is a family of vectors in a Hilbert space which allows
reproductions of arbitrary elements by continuous superpositions. Associated to
a given continuous frame we construct certain Banach spaces. Many classical
function spaces can be identified as such spaces. We provide a general method
to derive Banach frames and atomic decompositions for these Banach spaces by
sampling the continuous frame. This is done by generalizing the coorbit space
theory developed by Feichtinger and Groechenig. As an important tool the
concept of localization of frames is extended to continuous frames. As a
byproduct we give a partial answer to the question raised by Ali, Antoine and
Gazeau whether any continuous frame admits a corresponding discrete realization
generated by sampling.
