This booklet is devoted to one of the most important concepts of natural science, the concept of symmetry. The beneficial impact of the symmetry theory can be traced in the theory of elementary particles, crystallography, solid state physics, space and time theory, molecular biology, quantum chemistry, the study of arts, the theory of music as well as many branches of mathematics.Since many things are necessarily omitted even in a large monograph, this is all the more so in a small unpretentious booklet. Our narration does not cover transpositional symmetry, solid state theory and nu­ merous problems pertaining to application of symmetry theory to organic chemistry. This booklet deals pri­marily with spatial symmetry of molecules.The study of any theory is to a certain extent similar to the learning of a foreign language. While some peo­ ple learn it in order to write and speak fluently, others are satisfied with the understanding of foreign texts even if using a dictionary. The same is true in the case of a theory. While some people learn it very comprehen­sively in order to work actively in the area, it is suf­ficient for others to understand the language of theo­rists and the substance of their conclusions. This book is intended for the latter group, who are much more numerous than the former. This booklet is intended just for them. We address it to experimental chemists, teachers, university students and even high school senior students.The symmetry theory is usually applied to the specific problems of physics and chemistry when non- algebraic objects such as atoms, molecules, solids are studied with algebraic methods. Therefore some read­ers will probably need to overcome certain barriers of mathematical reasoning.The general logical plan of the booklet is as follows. The first chapter has a narrative nature and is dedicat­ed to the description of the major types of molecular symmetry. The second chapter may be considered as a mathematical model of the first. The next two chap­ ters are devoted to problems of the interrelation be­tween the composition, geometry and electronic structure of molecules (Chapter 3) and chemical reactions(Chapter 4). The final fifth chapter is of an historic and methodological character.The book was translated from the Russian by Yuti Atanov and was published by Mir in 1979.ContentsForeword 5Chapter One SYMMETRY OF A NUCLEAR POLYHEDRON 9Elements and operations of symmetry 9 Concept of a group 15 Hierarchy of point groups 19Chapter Two SYMMETRY IN MATHEMATICAL TERMS 30In the footsteps of Descartes 30Symmetry and molecular orbitals 42Chapter Three SYMMETRY AND STEREOCHEMISTRY 54 First variation on the theme of Coulomb law 55 Second variation on the theme of Coulomb law 63Theory of crystalline field (TCF) 71Electronic transitions 76Method of molecular orbitals and structure of complex compounds 81Jahn-Teller effect 92Pearson’s rule 96 Symmetry appears and disappears 104Chapter Four SYMMETRY AND CHEMICAL REACTION:  108A few words about chemical process 108 Permitting symmetry and forbidding symmetry 111 Orbital symmetry conservation in some organic reactions 114Chapter Five FORWARD INTO HISTORY!  125Σιμμετρια-Apocryphal times  125 Testament of Evariste Galois  126 Non-Euclidean geometries and chemistry 128 Symmetry concept in natural sciences  123 Group theory in quantum chemistry 142Index vocabulary  146