Abstract: Additive combinatorics is that branch of number theory concerned with finding arithmetic patterns in sets of integers. A beautiful example of a result in this subject is the Green-Tao theorem, that the prime numbers contain arbitrarily long sequences whose consectutive differences are all the same. While the statements of the theorems in additive combinatorics have a very strong number theory flavor, the proofs typically require tools from many subjects. One of the most fruitful tools is the discrete Fourier transform. In this talk, we will survey some of the basic results in additive combinatorics, emphasizing the role of Fourier analytic methods.