Given an irreducible polynomial f(x), is f(n) prime infinitely often? The conjectured answer, under some necessary conditions, is yes. However, we only know this to be true when f(x) is a linear polynomial and higher degree cases are considered notoriously difficult. In this talk, I will describe the recent trend of looking at an averaged version of this problem. In particular, I will discuss how when one averages over a large range of polynomial coefficients, one can achieve very precise information about the distribution of prime values of a polynomial (on average).