The problem of reconstructing a signal from measurements of the amplitude of its Fourier transform is a classic problem referred to as phase recovery and occurs in numerous applications in science and engineering. In this talk, we will assume that the underlying signal is sparse which is a very reasonable assumption in applications in astronomy, x-ray crystallography, wireless communications, etc. We first prove that, provided the support of the unknown signal is not periodic, it can be uniquely (up to time shifts and reflections) reconstructed from the magnitude of its Fourier transform. We then present a quadratic time algorithm that provides successful recovery, provided the sparsity is less $\sqrt{n}$, where n is the length of the unknown signal.