  Compressed sensing is a paradigm within signal processing that provides the
means for recovering structured signals from linear measurements in a highly
efficient manner. Originally devised for the recovery of sparse signals, it has
become clear that a similar methodology would also carry over to a wealth of
other classes of structured signals. In this work, we provide an overview over
the theory of compressed sensing for a particularly rich family of such
signals, namely those of hierarchically structured signals. Examples of such
signals are constituted by blocked vectors, with only few non-vanishing sparse
blocks. We present recovery algorithms based on efficient hierarchical
hard-thresholding. The algorithms are guaranteed to stable and robustly
converge to the correct solution provide the measurement map acts isometrically
restricted to the signal class. We then provide a series of results
establishing that the required condition for large classes of measurement
ensembles. Building upon this machinery, we sketch practical applications of
this framework in machine-type and quantum communication.
