Book excerpt: The Artin-Mazur zeta function of a dynamical system is a formal power series that enumerates the periodic points of all possible periods. This zeta function is well understood in characteristic 0 due to work of Artin, Mazur, Smale, Manning, Hinkkanen, and others. In particular, a rational function mapping the Riemann sphere P^1(C) to itself has a rational zeta function. This dissertation studies the algebraic structure of the zeta function for rational self-maps of P^1(k) for k a field of positive characteristic. For the family of dynamically affine rational maps, the question is completely resolved: it is shown that the zeta function is either rational or transcendental as a formal power series, and simple criteria are established to determine its rationality or transcendence. The proof proceeds by classifying all one-dimensional dynamically affine maps up to conjugacy, establishing explicit formulas to count periodic points, and using results from the theory of finite-state automata that control the algebraicity of power series.