Isaac Newton encrypted his discoveries in analysis in the form of an anagram that deciphers to the sentence, 'It is worthwhile to solve differential equations'. Accordingly, one can express the main idea behind the orbit method by saying 'It is worthwhile to study coadjoint orbits'. The orbit method was introduced by the author, A. A. Kirillov, in the 1960s and remains a useful and powerful tool in areas such as Lie theory, group representations, integrable systems, complex and symplectic geometry, and mathematical physics. This book describes the essence of the orbit method for non-experts and gives the first systematic, detailed, and self-contained exposition of the method. It starts with a convenient 'User's Guide' and contains numerous examples. It can be used as a text for a graduate course, as well as a handbook for non-experts and a reference book for research mathematicians and mathematical physicists.