Almost periodic solutions of differential equations have been studied since the very beginning of this century. The theory of almost periodic solutions has been developed in connection with problems of differential equations, dynamical systems, stability theory and its applications to control theory and other areas of mathematics. The classical books by C. Corduneanu [50], A.M. Fink [67], T. Yoshizawa [231], L. Amerio and G. Prouse [7], B.M. Levitan and V.V. Zhikov [137] gave a very nice presentation of methods as well as results in the area. In recent years, there has been an increasing interest in extending certain classical results to differential equations in Banach spaces. In this book we will make an attempt to gather systematically certain recent results in this direction.