The book presents a number of basic ideas and methods used to study ordinary differential equations. Elementary integration methods are considered from the point of view of general mathematical concepts (resolution of singularities, Lie groups of symmetries, Newton diagrams, etc.). The theory of first-order partial differential equations is expounded on the basis of contact structure geometry. The book includes classical and modern results of the theory of dynamical systems: structural stability, U-systems, analytical methods of local theory in the vicinity of a singular point or periodic solution (normal Poincaré forms), the theory of bifurcation of phase portraits when parameters change (soft and hard excitation of self-oscillations when stability is lost), doubling of the Feigenbaum period, Dulac's theorem, etc.