The sampling problem concerns the reconstruction of a function from samples (point evaluations) taken on a certain subset. The most classical results, due to Shannon and Beurling, involve bandlimited functions (that is, functions whose Fourier transform is supported on the unit interval) and provide a precise geometric characterization of all configurations of points that lead to reconstruction. I will present modern variants of these results and their implications in other fields, notably in the construction of functional expansions consisting of time-frequency shifts of a given profile function (known as Gabor or Weyl-Heisenberg systems).