The inverse problem of recovering a function from the magnitude of its Fourier transform is known as the phase retrieval problem and arises frequently in physics and engineering. Algorithms for phase retrieval have been in use for decades, however the convergence of these algorithms is often problematic or poorly understood. Recently, several novel approaches to the phase retrieval problem have been developed using techniques from mathematical optimization and computational harmonic analysis. As well, new variations of the phase retrieval problem have recently been identified that are key to emerging new technologies for molecular imaging. This talk will give an overview of current research on the phase retrieval problem.