We study the superposition operators (also called Nemytskii operators) between spaces of almost periodic functions with values in a complete metric space or in a Banach space. We establish new results on their continuity and their differentiability by using several different methods. We give applications to evolution equations and to differential equations in Banach spaces. We also establish new results on the superposition operators between spaces of asymptotically almost periodic functions and spaces of almost automorphic functions.