Several of the most important problems in combinatorial number theory ask for the size of the largest subset of an abelian group or interval of integers lacking points in some 'arithmetic' configuration. One example of such a question is, "What is the largest subset of {1,...,N} with no nontrivial k-term arithmetic progressions x,x+y,...,x+(k-1)y?". Gowers initiated the study of higher order Fourier analysis while seeking to answer this question and used it to give the first reasonable quantitative bounds. In this talk, I'll discuss what higher order Fourier analysis is and why it is relevant to the study of arithmetic progressions and other configurations, including 'polynomial' and 'multidimensional' configurations, and survey recent progress on problems related to the polynomial and multidimensional generalizations of Szemerédi's theorem. Speaker(s): Sarah Peluse (Princeton/IAS)