A signal vector can be easily reconstructed from a set of linear measurements such as the discrete Fourier transform. However, in many physical problems only the intensity, but not the phase, of the measurements can be obtained. From a mathematical perspective, the phase retrieval problem is to recover an unknown signal from the absolute values of a collection of measurements. This reconstruction problem has a long history in engineering and physics and arises in a variety of situations such as optics and speech recognition. In this talk we explain how, in many contexts, algebraic methods can be used to estimate the minimum number of phaseless measurements needed to recover generic signals.