  The problem of recovering a signal from its phaseless Fourier transform
measurements, called Fourier phase retrieval, arises in many applications in
engineering and science. Fourier phase retrieval poses fundamental theoretical
and algorithmic challenges. In general, there is no unique mapping between a
one-dimensional signal and its Fourier magnitude and therefore the problem is
ill-posed. Additionally, while almost all multidimensional signals are uniquely
mapped to their Fourier magnitude, the performance of existing algorithms is
generally not well-understood. In this chapter we survey methods to guarantee
uniqueness in Fourier phase retrieval. We then present different algorithmic
approaches to retrieve the signal in practice. We conclude by outlining some of
the main open questions in this field.
