| Definitions and examples | p. 1 |
| Preliminaries | p. 1 |
| Basic constructions | p. 4 |
| The projection method | p. 5 |
| The universal covering method | p. 8 |
| The suspension method | p. 6 |
| Whitney theorem | p. 11 |
| Connected sum of flows | p. 11 |
| The branch covering method | p. 12 |
| Basic examples | p. 12 |
| Gradient and Morse-Smale flows | p. 12 |
| Transitive flows | p. 13 |
| Flows with Cantor type limit sets | p. 14 |
| Area preserving and Hamiltonian flows | p. 17 |
| Harmonic and geodesic vector fields | p. 19 |
| Poincare-Bendixson's theory | p. 23 |
| Existence of closed transversal | p. 23 |
| Absence of non-trivial recurrent trajectories on some surfaces | p. 27 |
| Hilmy's and Cherry's theorems on quasiminimal sets | p. 28 |
| Maier's theorems on quasiminimal sets | p. 30 |
| Gutierrez's structure theorem | p. 34 |
| Limit set of individual trajectory | p. 35 |
| List of limit and minimal sets | p. 35 |
| Results of Solntzev and Vinograd | p. 37 |
| On the existence of minimal sets | p. 38 |
| Decomposition of flows | p. 41 |
| Decomposition theorems | p. 41 |
| Irreducible flows on torus | p. 41 |
| Canonical neighborhood | p. 42 |
| Gardiner - Levitt's decomposition | p. 42 |
| Pants decomposition | p. 45 |
| Decomposition of area preserving and Hamiltonian flows | p. 47 |
| Center of flow | p. 50 |
| Blowing-down of flows | p. 53 |
| Regular flows | p. 55 |
| Singular trajectories | p. 55 |
| Cells | p. 56 |
| Application: smoothing of flows | p. 59 |
| Local theory | p. 63 |
| Topological normal forms | p. 63 |
| Analytical normal forms | p. 64 |
| Smooth normal forms | p. 66 |
| Finitely smooth normal forms | p. 66 |
| Degenerate critical points | p. 67 |
| C1 normal forms of degenerate singularities | p. 70 |
| Space of flows and vector fields | p. 73 |
| Structural stability | p. 73 |
| Peixoto's graphs. Classification of Morse-Smale flows | p. 76 |
| Rotation systems | p. 76 |
| Peixoto theorems | p. 79 |
| Peixoto's counterexample revisited | p. 81 |
| Lyapunov's method | p. 83 |
| Lyapunov functions | p. 83 |
| Lyapunov graphs | p. 86 |
| Connected components of Morse-Smale flows | p. 87 |
| Degrees of non-stability | p. 89 |
| Typical properties of non-stable flows | p. 92 |
| Ergodic theory | p. 95 |
| Liouville's theorem | p. 96 |
| Kolmogorov's theorem for flows on torus | p. 98 |
| Non-trivial invariant measures | p. 100 |
| Ergodicity | p. 101 |
| Mixing | p. 106 |
| Entropy | p. 111 |
| Invariants of surface flows | p. 115 |
| Topological classification of torus flows | p. 116 |
| Rotation numbers | p. 116 |
| Classification of minimal flows | p. 119 |
| Classification of the Denjoy flows | p. 120 |
| Classification of flows of the Cherry type | p. 121 |
| Oriented surfaces of genus ≥ 2 | p. 128 |
| Aranson-Grines homotopy rotation class | p. 125 |
| Homotopy rotation orbit | p. 126 |
| Equivalence of irrational flows | p. 127 |
| Properties of the homotopy rotation classes | p. 128 |
| Application of geodesic laminations | p. 129 |
| Transitive flows on non-orientable surfaces | p. 132 |
| Torus with a cross-cup | p. 132 |
| Non-orientable surfaces of genus ≥ 4 | p. 134 |
| Classification of exceptional minimal sets | p. 134 |
| Classification of the regular flows | p. 135 |
| Leontovich-Maier's theorem for sphere flows | p. 136 |
| Neumann-O'Brien's orbit complex | p. 136 |
| Bolsinov-Fomenko's classification of Hamiltonian flows | p. 139 |
| Classification of non-wandering flows | p. 141 |
| Elementary cells of non-wandering flows | p. 142 |
| Conley-Lyapunov-Peixoto graphs | p. 144 |
| Equivalence Problem | p. 145 |
| Realization Problem | p. 146 |
| Cayley graph of a flow | p. 147 |
| Finite groups and Cayley graphs | p. 147 |
| Isomorphism Problem | p. 149 |
| Realization Problem | p. 150 |
| Homology and cohomology invariants | p. 151 |
| Asymptotic cycles | p. 152 |
| Fundamental class of A. Katok | p. 155 |
| Zorich's cycles | p. 160 |
| Rotation sets of surface flows | p. 163 |
| Smooth classification of flows | p. 164 |
| Torus and Klein bottle | p. 165 |
| Closed orientable surfaces of genus ≥ 2 | p. 167 |
| C*-algebras of surface flows | p. 175 |
| Irrational rotation algebra | p. 175 |
| Dimension groups | p. 176 |
| Continued fractions | p. 178 |
| Effros-Shen's Theorem | p. 179 |
| Embedding of A¿ | p. 180 |
| Projections of A¿ | p. 181 |
| Morita Equivalence | p. 183 |
| Artin's rotation algebra | p. 184 |
| Myrberg's Approximationssatz | p. 184 |
| Artin's numbers | p. 186 |
| K-theory | p. 193 |
| Torus with Reeb's components | p. 195 |
| Baum-Connes Conjecture | p. 195 |
| C*-algebras of Morse-Smale flows | p. 197 |
| Semi-local theory | p. 201 |
| Denjoy's and Schwarz's theorems | p. 202 |
| Cherry's problem | p. 203 |
| Local structure preventing quasiminimality | p. 207 |
| Anosov-Weil problem | p. 209 |
| Theorems of Weil and Anosov | p. 210 |
| Asymptotic directions | p. 210 |
| Weil's theorem and Weil's conjecture | p. 211 |
| Anosov's theorem | p. 213 |
| Proof of Weil's conjecture and Weil's theorem | p. 216 |
| Asymptotic direction of individual curves | p. 217 |
| Non-trivial recurrent semi-trajectories | p. 217 |
| Trajectories of analytic flows | p. 219 |
| Leaves of foliation | p. 219 |
| Curves with restriction on the geodesic curvature | p. 220 |
| Approximation of curve by trajectories of a flow | p. 221 |
| Limit sets of curves and trajectories at the absolute | p. 222 |
| Deviation of curves from the geodesies | p. 224 |
| The deviation property of trajectories | p. 224 |
| Deviation from the geodesic frameworks | p. 227 |
| Branched coverings | p. 228 |
| Swing of trajectories near hyperbolic lines | p. 229 |
| Examples of unbounded deviation | p. 233 |
| Surfaces of genus ≥ 2 | p. 233 |
| Irrational direction on torus | p. 235 |
| Rational direction on torus | p. 236 |
| Non-compact surfaces | p. 239 |
| Kaplan's classification | p. 239 |
| Level curves of harmonic functions | p. 242 |
| Markus's classification | p. 245 |
| Structural stability | p. 246 |
| Neumann's example | p. 248 |
| Inaba's example and Beniere-Meigniez's theorem | p. 250 |
| Beniere-Hector's theorem | p. 252 |
| Aranson-Zhuzhoma's example | p. 253 |
| Triptych | p. 257 |
| Geodesic frameworks revisited | p. 257 |
| On continuity and collapse of geodesic frameworks | p. 260 |
| Cr-closing lemma | p. 263 |
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