In this book we give an extensive overview of univariate and multivariate weighted renewal functions. Such functions are closely related to ordinary renewal functions and processes and have been investigated by many authors. Our main goal is to bring together a number of old and new results and to provide a uniform approach. The material presented in the book is intended for graduate students and researchers in mathematics, probability theory and stochastic processes. In the proofs of our results we use probability arguments as well as Abelian and Tauberian results for the Laplace transform of functions and sequences. A pre-requisite for this book is an introduction to probability theory and stochastic processes. In the appendix of the book we give a summary of the theory of regularly varying functions and related classes of functions.