In contemplating the third edition, I have had multiple objectives to achieve. The ?rst and foremost important objective is to maintain the - cessibility and readability of the book to a broad readership with varying mathematical backgrounds and sophistication. More proofs, more graphs, more explanations, and more applications are provided in this edition. The second objective is to update the contents of the book so that the reader stays abreast of new developments in this vital area of mathematics. Recent results on local and global stability of one-dimensional maps are included in Chapters 1, 4, and Appendices A and C. An extension of the Hartman?Grobman Theorem to noninvertible maps is stated in Appendix D. A whole new section on various notions of the asymptoticity of solutions and a recent extension of Perron?s Second Theorem are added to Chapter 8. In Appendix E a detailed proof of the Levin?May Theorem is presented. In Chapters 4 and 5, the reader will ?nd the latest results on the larval? pupal?adult ?our beetle model. The third and ?nal objective is to better serve the broad readership of this book by including most, but certainly not all, of the research areas in di?erence equations. As more work is being published in the Journal of Di?erence Equations and Applications and elsewhere, it became apparent that a whole chapter needed to be dedicated to this enterprise. With the prodding and encouragement of Gerry Ladas, the new Chapter 5 was born.