The super-resolution problem is to recover a unknown measure from knowledge of its perturbed low-frequency Fourier coefficients. When the measure is discrete and the minimum distance between its Dirac masses is below the Rayleigh length, the problem is especially difficult, but it is relevant to many imaging and signal processing tasks. We introduce both a discrete and a continuous super-resolution model. For the discrete model, we provide a sharp bound for the min-max recovery error by establishing a sharp bound on the smallest singular value of restricted Fourier matrices. For the continuous model, we study the total variation minimization method and borrow ideas from Beurling in order to describe its solutions without additional conditions on the measure and sampling set. This presentation includes joint work with John J. Benedetto and Wenjing Liao.