This volume is a systematic treatment of the additive number theory of polynomials over a finite field, an area possessing deep and fascinating parallels with classical number theory. In providing asymptomatic proofs of both the Polynomial Three Primes Problem (an analog of Vinogradov's theorem) and the Polynomial Waring Problem, the book develops the various tools necessary to apply an adelic "circle method" to a wide variety of additive problems in both the polynomial and classical settings. A key to the methods employed here is that the generalized Riemann hypothesis is valid in this polynomial setting. The authors presuppose a familiarity with algebra and number theory as might be gained from the first two years of graduate course, but otherwise the book is self-contained. Starting with analysis on local fields, the main technical results are all proved in detail so that there are extensive discussions of the theory of characters in a non-Archimidean field, adele class groups, the global singular series and Radon-Nikodyn derivatives, L-functions of Dirichlet type, and K-ideles.