In a celebrated paper from several decades ago, Donoho and Stark laid down the foundations for studying signal recovery using the Fourier uncertainty principle. We are going to describe these ideas and discuss how the Fourier restriction phenomenon, extensively studied in harmonic analysis in a variety of contexts, can be brought to bear on results of this type. We are going to begin by reviewing the basics of the restriction phenomenon, reformulate it in terms of signal recovery, and then move on to the discrete setting. This is joint work with Azita Mayeli from the City University of New York Graduate Center.