The main purpose of this book is to present some of the old and recent results on Hyers-Ulam stability of function equations and differential equations in Banach spaces, quasi-Banach spaces, F-spaces and group. The book provides a survey of both the latest and new results especially on the following topics: (1)Stability theory for several new functional equations in Banach spaces. (2)Stability of some functional equations and some properties of groups. (3)Hyers-Ulam Stability of functional differential equations and partial differential equations. (4)Stability of nonlinear fractional differential equations. (5)Stability of differential equations on time scales