The main purpose of this book is to introduce the generalized composition operators and the generalized multiplication operators on weighted Hardy spaces . Composition operators have been the subject matter of intensive study over past several decades and they find applications in ergodic theory, quantum mechanics and other branches of mathematical and physical sciences . The 1st chapter of the book contains introductory material . The compact, Fredholm generalized composition operators are characterized in the second chapter. The Hermitian, Normal, Quasi normal, Isometric and co isometric generalized composition operators are presented in the third chapter. It is shown that most of the properties of the generalized multiplication operators differ from the multiplication operators acting on other function spaces .eg: - On most of the function spaces multiplication operators are normal .But this result is not valid incase of generalized multiplication operators on weighted Hardy spaces . In this book some light is thrown on differential and anti-differential operators acting on weighted Hardy spaces .