Abstract: We consider the abstract problem of recovering an unknown function from a finite number of (potentially noisy) measurements. Given different assumptions on the target function, different error measures, and different restrictions on the measurements allowed, this leads to a variety of interesting problems. We will survey some of the known results on these types of questions, discuss their practical implications, and pose open problems and further lines of research. A specific problem we will focus on to highlight these results is the question of how to optimally recover functions from a Besov space using samples of the Fourier transform, with error measured in the Lp norm.