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Classically, discrete dynamics is the study of iteration of polynomial and rational maps on the complex plane or on the Riemann sphere. The arithmetic properties for the associated dynamical system were studied by J. Silverman in his paper "Integer points, Diophantine approximation, and iteration of rational maps" (1993). Motivated by the analogy between Nevanlinna theory and Diophantine approximation discovered by C. F. Osgood and P. Vojta, M. Ru and E. Yi studied the complex hyperbolic properties for the associated dynamical systems in their paper "Nevanlinna Theory and Iteration of Rational Maps" (2005). Finiteness in Diophantine approximation corresponds to constancy of a meromorphic function in Nevanlinna Theory. This book discusses the iteration of rational maps in relation to Diophantine approximation, as well as Nevanlinna theory, based on the above mentioned papers.