This book is the result of a 20 year research study carried out by the author in the fields of Theoretical Continuum Mechanics, Finite Element Method (FEM) and Numerical Analysis, devoted to develop a single powerful numerical tool for the solution of non-linear time dependent Continuum Mechanics problems.This book addresses people who are involved in the engineering science and want to learn about the nature and the capabilities of the FEM.The author has attempted to make the book self-contained and comprehensive enough to meet the needs of students as well as those of most practising engineers and scientists.The first part of the book is conceived to be a complete overview on the FEM and it can be used as a textbook for an introductory course.The second part is devoted to an in-depth investigation of the FEM when applied to the different engineering fields and it can therefore be used as a textbook for an advanced course.