Fourier introduced the idea that every periodic function can be represented as an exponential sum. There have been a lot of efforts to understand exponential sums in algebraic geometry, additive combinatorics, number theory, and Fourier analysis. Although a group of people from different areas tried to have a better understanding of exponential sums, it is surprising to me that basic questions for exponential sums still remain open. I'll introduce some backgrounds on exponential sums and present a recent work with Larry Guth and Dominique Maldague.