We discuss the problem of bounding the Fourier transforms of stationary measures of iterated function systems (IFSs) and how the pseudo-randomness of the IFS either due to arithmetic, algebraic or geometric reasons is reflected in the behaviour of the Fourier transform. We outline various methods that have been built to estimate the Fourier transform of stationary measures arising e.g. from thermodynamical formalism, additive combinatorics, random walks on groups and hyperbolic dynamics. Open problems, prospects and recent links to quantum chaos are also highlighted.