Understanding the arithmetic of sequences that are dynamically defined, such as the Fibonacci and Mersenne numbers, is of classical interest but generally results are difficult; for example, the infinitude of primes in either of these sequences is still open. However, results on the existence of primitive prime divisors and perfect powers in these sequences have been achieved, including the notable theorem of Bugeaud, Mignotte, and Siksek listing the Fibonacci powers. These questions and methods generalize to sequences which are forward orbits under iteration of certain dynamical systems, and I will discuss results on the arithmetic of such sequences, which rely on techniques from Diophantine approximation, arithmetic dynamics, and complex dynamics.