Phase retrieval is one of the classical problems in various fields of physics including x-ray crystallography, astronomy and spectroscopy. It arises when only an amplitude measurement on electric field can be made while both amplitude and phase of the field are needed for obtaining the desired material properties. In optical and terahertz spectroscopies, in particular, phase retrieval is a one-dimensional problem, which is considered as unsolvable in general. Nevertheless, an approach utilizing the maximum entropy principle has proven to be a feasible tool in various applications of optical, both linear and nonlinear, as well as in terahertz spectroscopies, where the one-dimensional phase retrieval problem arises. In this review, we focus on phase retrieval using the maximum entropy method in various spectroscopic applications. We review the theory behind the method and illustrate through examples why and how the method works, as well as discuss its limitations.