Definitions and examples -- Poincaré-Bendixson’s theory -- Decomposition of flows -- Local theory -- Space of flows and vector fields -- Ergodic theory -- Invariants of surface flows -- C *-algebras of surface flows -- Semi-local theory -- Anosov-Weil problem -- Non-compact surfaces -- Triptych.

Time-evolution in low-dimensional topological spaces is a subject of puzzling vitality. This book is a state-of-the-art account, covering classical and new results. The volume comprises Poincaré-Bendixson, local and Morse-Smale theories, as well as a carefully written chapter on the invariants of surface flows. Of particular interest are chapters on the Anosov-Weil problem, C*-algebras and non-compact surfaces. The book invites graduate students and non-specialists to a fascinating realm of research. It is a valuable source of reference to the specialists.