  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 converge, in a stable
fashion both with respect to measurement noise as well as to model mismatches,
to the correct solution provided the measurement map acts isometrically
restricted to the signal class. We then provide a series of results
establishing the required condition for large classes of measurement ensembles.
Building upon this machinery, we sketch practical applications of this
framework in machine-type communications and quantum tomography.
