For the first English edition (1983) see Zbl 0507.34003. In the time interval between the two editions there was substantial progress in the subject studied in the book. In the second edition some new material representing various aspects of this progress has been added, as the author points out in the preface. Naturally new references are added to the original bibliography; unfortunately there is still no bibliographic index at the end of the book. We found it best to include the author’s preface in this review. From the author’s preface: “Since 1978, when the first Russian edition of this book appeared, geometrical methods in the theory of ordinary differential equations have become very popular. A lot of computer experiments have been performed and some theorems have been proved. In this edition, this progress is (partially) represented by some additions to the first English text. I mention here some of these recent discoveries. 1. The Feigenbaum universality of period doubling cascades and its extensions - the renormalization group analysis of bifurcations (Percival, Landford, Sinai,...). 2. The Zoladek solution of the two- parameter bifurcation problem (cases of two imaginary pairs of eigenvalues and of a zero eigenvalue and a pair). 3. The Iljashenko proof of the “Dulac theorem” on the finiteness of the number of limit cycles of polynomial planar vector fields. 4. The Ecalle and Veronin theory of holomorphic invariants for formally equivalent dynamical systems at resonances. 5. The Varchenko and Hovanski theorems on the finiteness of the number of limit cycles generated by a polynomial perturbation of a polynomial Hamiltonian system (the Dulac form of the weakened version of Hilbert’s sixteenth problem). 6. The Petrov estimates of the number of zeros of the elliptic integrals responsible for the birth of limit cycles for polynomial perturbations of the Hamiltonian system - 1 (solution of the weakened sixteenth Hilbert problem for cubic Hamiltonians). 7. The Bachtin theorems on averaging in systems with several frequencies. 8. The Davydov theory of normal forms for singularities of implicit differential equations and relaxation oscillations. 9. The Neistadt and Cary-Escande- Tennyson theory of adiabatic invariant’s change under separatrix crossing (explaining, according to Wisdom, the Kirkwood gaps in the distribution of asteroids). 10. The Neistadt theory of dynamical bifurcations. The problem of bifurcations at 1:4 resonance seems to be still unsolved, but present the conjectural answer supported by both computer experiments and asymptotic analysis. I mention here some other important recent results: (1) the bifurcation theory of fundamental systems of solutions of linear equations (related to the Schubert stratification of the Grassmannians and to the Weierstrass points on algebraic curves) by M. Kazarian; (2) the theory of normal forms of vector fields with Jordan linear part (related to sl(2)-modules) by Bogaevski, Povzner, and Giventhal; (3) the bifurcation theory of cycles in reversible systems (related to the metamorphoses of the umbrella’s section) by M. Sevrjuk; (4) the theory of nonoscillatory linear equation (related to the geometry of the Schubert stratification of the flag manifolds); (5) the classification of the local topological bifurcations in generic gradient systems depending on three parameters (related to the Thom conjecture on catastrophes) by B. Hessin; (6) the theory of versal deformations for the vector fields on a line (related to the differential forms of complex degree) by V. Kostov; (7) the tunelling asymptotics in systems with many competing attractors (related the Fokker-Planck equation and to the Witten inequalities) by V. Fok; (8) the bifurcation theory for planar homogeneous vector fields (related to the higher dimensional umbrellas) by B. Hessin.