A Fourier quasicrystal is a discrete subset of Euclidean space whose Fourier transform is also discrete. These sets are almost periodic, often without being periodic or even containing long arithmetic progressions. In 2020, Kurasov and Sarnak gave a method for constructing one-dimensional Fourier quasicrystals from multivariate stable polynomials, which are polynomials whose zeroes avoid certain regions in the complex plane. The rich theory of stable polynomials can be used to control and extract features of the corresponding quasicrystals. Analogous varieties of higher codimension also give rise to higher dimensional quasicrystals. I will give an introduction to these objects and the beautiful connection between them. This is based on joint works with Lior Alon, Alex Cohen, Mario Kummer, and Pavel Kurasov.