This book presents the subject of integral equations in an accessible manner for a variety of applications. Emphasis is placed on understanding the subject while avoiding the abstract and compact theorems. A feature of the book is that it introduces the powerful and reliable developments in this field, not covered in traditional texts. The newly developed decomposition method, the series solution method and the direct computation method are thoroughly implemented, which allows the topic to be far more accessible. The book also includes some of the traditional techniques for comparison. Using the newly developed methods, the author covers Fredholm and Volterra integral equations, singular integral equations, integro-differential equations and nonlinear integral equations, with promising results for linear and nonlinear models. Many examples are given to introduce the material. In addition, many exercises are provided to build confidence, ease and skill in using the new methods. This book may be used as a text for advanced undergraduates and graduate students in mathematics and scientific areas, and as a work of reference for research study of differential equations and numerical analysis.