Over the past two decades, the method of fundamental solutions (MFS) has attracted great attention and has been used extensively for the solution of scientific and engineering problems. The MFS is a boundary meshless collocation method which has evolved from the boundary element method. In it, the approximate solution is expressed as a linear combination of fundamental solutions of the operator in the governing partial differential equation.One of the main attractions of the MFS is the simplicity with which it can be applied to the solution of boundary value problems in complex geometries in two and three dimensions. The method is also known by many different names in the literature such as the charge simulation method, the de-singularization method, the virtual boundary element method, etc.Despite its effectiveness, the original version of the MFS is confined to solving boundary value problems governed by homogeneous partial differential equations. To address this limitation, we introduce various types of particular solutions to extend the method to solving general inhomogeneous boundary value problems employing the method of particular solutions.This book consists of two parts. Part I aims to provide theoretical support for beginners. In the spirit of reproducible research and to facilitate the understanding of the method and its implementation, several MATLAB codes have been included in Part II.This book is highly recommended for use by post-graduate researchers and graduate students in scientific computing and engineering.