Compressive sensing is a novel theory which asserts that one can recover signals or images of interest with far fewer measurements or data bits than were thought necessary. The first part of the talk will introduce some of the theory and survey important applications which allow -- among other things -- faster and cheaper imaging. For instance, compressive sensing asserts that under sparsity constraints, one can recover or interpolate the whole spectrum of an object exactly from just a few randomly spaced samples by solving a simple convex program. In many applications, however, we cannot sample the spectrum at random locations; rather, one can only observe low-frequencies as there usually is a physical limit on the highest possible resolution. Is it then possible to extrapolate the spectrum and recover the high-frequency band? The second part of the talk will introduce recent results towards a mathematical theory of super-resolution -- a word used in different contexts mainly to design techniques for enhancing the resolution of a sensing system.