  This is a survey of the use of Fourier analysis in additive combinatorics,
with a particular focus on situations where it cannot be straightforwardly
applied, but needs to be generalized first. Sometimes very satisfactory
generalizations exist, while sometimes we have to make do with theories that
have some of the desirable properties of Fourier analysis but not all of them.
In the latter case, there are intriguing hints that there may be more
satisfactory theories yet to be discovered. This article grew out of the
Colloquium Lectures at the Joint Meeting of the AMS and the MAA, given in
Seattle in January 2016.
