It is surprisingly straightforward to count the number of solutions to simple equations such as x+y=2z or x-y=z^2, where x, y and z lie in a "random-looking" subset of the integers. The discrete Fourier transform provides a natural way of quantifying what we mean by random-looking, but fails us once we start to consider arithmetic progressions of length greater than three and other more sophisticated structures. This failure opens the door to a rich and still evolving theory of higher-degree Fourier analysis, which we shall try and catch a glimpse of in this talk. This talk aims to be accessible to postgraduate students across all areas of mathematics.