This chapter provides an overview of the book "Dichotomies in Stability Theory" by W. A. Coppel, published by Springer-Verlag in 1978. The book is part of the Lecture Notes in Mathematics series and is edited by A. Dold and B. Eckmann. It is based on a course given at the University of Florence in May 1977. The author, W. A. Coppel, emphasizes his belief that dichotomies are crucial for understanding the asymptotic behavior of non-autonomous differential equations, despite the recent focus on characteristic exponents. The book covers various aspects of dichotomies, including exponential and ordinary dichotomies, their relationship with functional analysis, roughness, reducibility, criteria for exponential dichotomies, and their connection to Lyapunov functions. It also discusses equations on R and almost periodic equations, as well as the hull of an equation. The content is structured into nine lectures, each focusing on a specific topic, and includes an appendix on the Perron method, notes, references, and a subject index.This chapter provides an overview of the book "Dichotomies in Stability Theory" by W. A. Coppel, published by Springer-Verlag in 1978. The book is part of the Lecture Notes in Mathematics series and is edited by A. Dold and B. Eckmann. It is based on a course given at the University of Florence in May 1977. The author, W. A. Coppel, emphasizes his belief that dichotomies are crucial for understanding the asymptotic behavior of non-autonomous differential equations, despite the recent focus on characteristic exponents. The book covers various aspects of dichotomies, including exponential and ordinary dichotomies, their relationship with functional analysis, roughness, reducibility, criteria for exponential dichotomies, and their connection to Lyapunov functions. It also discusses equations on R and almost periodic equations, as well as the hull of an equation. The content is structured into nine lectures, each focusing on a specific topic, and includes an appendix on the Perron method, notes, references, and a subject index.