One important technical ingredient in many arithmetic statistics papers is upper bounds for finite exponential sums which arise as Fourier transforms of characteristic functions of orbits. This is typical in results obtaining power saving error terms, treating ``local conditions'', and/or applying any sort of sieve. In my talk I will explain what these exponential sums are, how they arise, and what their relevance is. I will outline a new method for explicitly and easily evaluating them, and describe some pleasant surprises in our end results. I will also outline a new sieve method for efficiently exploiting these results, involving Poisson summation and the Bhargava-Ekedahl geometric sieve. For example, we have proved that there are ``many'' quartic field discriminants with at most eight prime factors. This is joint work with Takashi Taniguchi.