The geometry of periodic points is a fundamental feature of a dynamical system. In an algebraic setting, where the system is defined by polynomials, we can use tools from algebraic or arithmetic geometry (or model theory) to study the structure of these finite orbits. Important examples come from the study of abelian varieties, but already the setting of polynomials of one variable is a challenge. In this talk, I will give an overview of results from the last 30 years that address the intersection of Preper(f) and Preper(g), the sets of preperiodic points for two maps f and g on P^1. Very recently, we completed one line of inquiry by showing there is a uniform bound on the size of the intersection, depending only on the degree and assuming the two sets are not equal.