  The problem of phase retrieval, i.e., the problem of recovering a function
from the magnitudes of its Fourier transform, naturally arises in various
fields of physics, such as astronomy, radar, speech recognition, quantum
mechanics and, perhaps most prominently, diffraction imaging. The mathematical
study of phase retrieval problems possesses a long history with a number of
beautiful and deep results drawing from different mathematical fields, such as
harmonic analyis, complex analysis, or Riemannian geometry. The present paper
aims to present a summary of some of these results with an emphasis on recent
activities. In particular we aim to summarize our current understanding of
uniqueness and stability properties of phase retrieval problems.
