  Consider a scenario in which an unknown signal is transformed by a known
linear operator, and then the pointwise absolute value of the unknown output
function is reported. This scenario appears in several applications, and the
goal is to recover the unknown signal -- this is called phase retrieval. Phase
retrieval has been a popular subject of research in the last few years, both in
determining whether complete information is available with a given linear
operator, and in finding efficient and stable phase retrieval algorithms in the
cases where complete information is available. Interestingly, there are a few
ways to measure information completeness, and each way appears to be governed
by a phase transition of sorts. This chapter will survey the state of the art
with some of these phase transitions, and identify a few open problems for
further research.
