This is a lecture note on dichotomies in stability theory, written by W. A. Coppel. The notes are based on a course given at the University of Florence in May 1977. The author expresses the view that dichotomies, rather than Lyapunov's characteristic exponents, are key to understanding the asymptotic behavior of non-autonomous differential equations. Despite the increased number of papers and a book on characteristic exponents, the author believes dichotomies remain important. There has been recent development in the theory of dichotomies, making it an appropriate time to present an accessible account of the theory. The lecture notes cover various topics, including stability, exponential and ordinary dichotomies, dichotomies and functional analysis, roughness, dichotomies and reducibility, criteria for an exponential dichotomy, dichotomies and Lyapunov functions, equations on R and almost periodic equations, dichotomies and the hull of an equation, and an appendix on the method of Perron. The notes also include a list of references and a subject index. The work is published by Springer-Verlag in 1978. The author thanks Professor R. Conti for inviting him to visit the University of Florence and for encouraging him to organize his thoughts. He also thanks Mrs. Helen Daish and Mrs. Linda Southwell for their help in typing the manuscript. The book is subject to copyright and may be reproduced with the permission of the publisher.This is a lecture note on dichotomies in stability theory, written by W. A. Coppel. The notes are based on a course given at the University of Florence in May 1977. The author expresses the view that dichotomies, rather than Lyapunov's characteristic exponents, are key to understanding the asymptotic behavior of non-autonomous differential equations. Despite the increased number of papers and a book on characteristic exponents, the author believes dichotomies remain important. There has been recent development in the theory of dichotomies, making it an appropriate time to present an accessible account of the theory. The lecture notes cover various topics, including stability, exponential and ordinary dichotomies, dichotomies and functional analysis, roughness, dichotomies and reducibility, criteria for an exponential dichotomy, dichotomies and Lyapunov functions, equations on R and almost periodic equations, dichotomies and the hull of an equation, and an appendix on the method of Perron. The notes also include a list of references and a subject index. The work is published by Springer-Verlag in 1978. The author thanks Professor R. Conti for inviting him to visit the University of Florence and for encouraging him to organize his thoughts. He also thanks Mrs. Helen Daish and Mrs. Linda Southwell for their help in typing the manuscript. The book is subject to copyright and may be reproduced with the permission of the publisher.