This book is an introduction to the qualitative theory of dynamical systems on manifolds of low dimension (on the circle and on surfaces). Along with classical results, it reflects the most significant achievements in this area obtained in recent times by Russian and foreign mathematicians whose work has not yet appeared in the monographic literature. The main stress here is put on global problems in the qualitative theory of flows on surfaces. Despite the fact that flows on surfaces have the same local structure as flows on the plane, they have many global properties intrinsic to multidimensional systems. This is connected mainly with the existence of nontrivial recurrent trajectories for such flows. The investigation of dynamical systems on surfaces is therefore a natural stage in the transition to multidimensional dynamical systems. The reader of this book need be familiar only with basic courses in differential equations and smooth manifolds. All the main definitions and concepts required for understanding the contents are given in the text. The results expounded can be used for investigating mathematical models of mechanical, physical, and other systems (billiards in polygons, the dynamics of a spinning top with nonholonomic constraints, the structure of liquid crystals, etc.). In our opinion the book should be useful not only to mathematicians in all areas, but also to specialists with a mathematical background who are studying dynamical processes: mechanical engineers, physicists, biologists, and so on. Readership Graduate students and researchers working in dynamical systems and differential equations, as well as specialists with a mathematical background who are studying dynamical processes: mechanical engineers, physicists, biologists, etc. Reviews "These and many other wonders are revealed in this thorough monograph. Lovers of dynamical systems will find this a mine of interesting information."  Bulletin of the London Mathematical Society "Consists of seven wellwritten chapters with mathematical rigor, and only prerequisite knowledge of topology and differential equations on the level of undergraduate students is assumed ... contains ... not only rich material for studying dynamical systems of twodimensional manifolds, but also a natural background for understanding properties of multidimensional dynamical systems."  Zentralblatt MATH "Comprehensive ... serves as a good reference for flows on surfaces, and would be well suited for a specialized graduate course on these topics ... very well written."  Mathematical Reviews Table of Contents  Dynamical systems on surfaces
 Structure of limit sets
 Topological structure of flows
 Local structure of dynamical systems
 Transformations of the circle
 Classification of flows on surfaces
 Relation between smoothness properties and topological properties of flows
 Bibliography
