We propose a general framework to perform statistical estimation from low-resolution data, a crucial challenge in many applications. First, we show that solving a simple convex program allows to super-resolve a superposition of point sources from bandlimited measurements with infinite precision. This holds as long as the sources are separated by a distance related to the cut-off frequency of the data. The result extends to higher dimensions and to the super-resolution of piecewise-smooth functions. Then, we provide theoretical guarantees that establish the robustness of our methods to noise in a non-asymptotic regime. Finally, we illustrate the flexibility of the framework by discussing extensions to the demixing of sines and spikes and to super-resolution from multiple measurements.