This talk is concerned with the inverse problem of recovering a discrete measure on the torus consisting of S atoms, given M consecutive noisy Fourier coefficients. Super-resolution is sensitive to noise when the distance between two atoms is less than 1/M. We connect this problem to the minimum singular value of non-harmonic Fourier matrices. New results for the latter are presented, and as consequences, we derive results regarding the information theoretic limit of super-resolution and the resolution limit of subspace methods (namely, MUSIC and ESPRIT). These results rigorously establish the super-resolution phenomena of these algorithms that were empirically discovered long ago, and numerical results indicate that our bounds are sharp or nearly sharp. We also discuss how to take advantage of redundant measurements for the purpose of reducing quantization error. Interesting connections to trigonometric interpolation and uncertainty principles are also presented. Joint work with John Benedetto, Albert Fannjiang, Sinan Gunturk, and Wenjing Liao.