As David Speyer discussed last week, hyperplane arrangements were a major motivation for the study of matroids. In this talk, we’ll take a more in-depth look at hyperplane arrangements with a focus on their characteristic polynomials. While initially defined in terms the point count over F_q, the characteristic polynomial can be computed entirely combinatorially from the intersection poset. Moreover, in certain situations, the characteristic polynomial has a physical interpretation; over R, it gives the number of connected components and, over C, it gives the Betti numbers. This talk will highlight these results and some of the techniques used to prove them.