This comprehensive textbook is designed to take undergraduate and beginning graduate students of mathematics, science, and engineering from the rudimentary beginnings to the exciting frontiers of dynamical systems and their applications. It is a masterful exposition of the foundations of ordinary differential and difference equations from the contemporary viewpoint of dynamical systems and bifurcations. In both conception and execution, the authors implemented a fresh approach to mathematical narration. Fundamental ideas are explained in simple settings, the ramifications of theorems are explored for specific equations, and above all, the subject is related in the guise of a mathematical epic. With its insightful and engaging style, as well as its numerous computer-drawn illustrations of notable equations of theoretical and practical importance, this unique book will simply captivate the attention of students and instructors alike. 345 illustrations.