1. Classical two-dimensional oscillating systems and their multidimensional analogues -- ยง1.1. The van der Pol equation -- ยง1.2. The equation of oscillations of a pendulum -- ยง1.3. Oscillations in two-dimensional systems with hysteresis -- ยง1.4. Lower estimates of the number of cycles of a two-dimensional system -- 2. Frequency criteria for stability and properties of solutions of special matrix inequalities -- ยง2.1. Frequency criteria for stability and dichotomy -- ยง2.2. Theorems on solvability and properties of special matrix inequalities -- 3. Multidimensional analogues of the van der Pol equation -- ยง3.1. Dissipative systems. Frequency criteria for dissipativity -- ยง3.2. Second-order systems. Frequency realization of the annulus principle -- ยง3.3. Third-order systems. The torus principle -- ยง3.4. The main ideas of applying frequency methods for multidimensional systems -- ยง3.5. The criterion for the existence of a periodic solution in a system with tachometric feedback -- ยง3.6. The method of transition into the "space of derivatives" -- ยง3.7. A positively invariant torus and the function "quadratic form plus integral of nonlinearity" -- ยง3.8. The generalized Poincarรฉโ{128}{147}Bendixson principle -- ยง3.9. A frequency realization of the generalized Poincarรฉ-Bendixson principle -- ยง3.10. Frequency estimates of the period of a cycle -- 4. Yakubovich autoโ{128}{147}oscillation -- ยง4.1. Frequency criteria for oscillation of systems with one differentiable nonlinearity -- ยง4.2. Examples of oscillatory systems -- 5. Cycles in systems with cylindrical phase space -- ยง5.1. The simplest case of application of the nonlocal reduction method for the equation of a synchronous machine -- ยง5.2. Circular motions and cycles of the second kind in systems with one nonlinearity -- ยง5.3. The method of systems of comparison -- ยง5.4. Examples -- ยง5.5. Frequency criteria for the existence of cycles of the second kind in systems with several nonlinearities -- ยง5.6. Estimation of the period of cycles of the second kind -- 6. The Barbashin-Ezeilo problem -- ยง6.1. The existence of cycles of the second kind -- ยง6.2. Bakaev stability. The method of invariant conical grids -- ยง6.3. The existence of cycles of the first kind in phase systems -- ยง6.4. A criterion for the existence of nontrivial periodic solutions of a third-order nonlinear system -- 7. Oscillations in systems satisfying generalized Routh-Hurwitz conditions. Aizerman conjecture -- ยง7.1. The existence of periodic solutions of systems with nonlinearity from a Hurwitzian sector -- ยง7.2. Necessary conditions for global stability in the critical case of two zero roots -- ยง7.3. Lemmas on estimates of solutions in the critical case of one zero root -- ยง7.4. Necessary conditions for absolute stability of nonautonomous systems -- ยง7.5. The existence of oscillatory and periodic solutions of systems with hysteretic nonlinearities -- 8. Frequency estimates of the Hausdorff dimension of attractors and orbital stability of cycles -- ยง8.1. Upper estimates of the Hausdorff measure of compact sets under differentiable mappings -- ยง8.2. Estimate of the Hausdorff dimension of attractors of systems of differential equations -- ยง8.3. Global asymptotic stability of autonomous systems -- ยง8.4. Zhukovsky stability of trajectories -- ยง8.5. A frequency criterion for Poincarรฉ stability of cycles of the second kind -- ยง8.6. Frequency estimates for the Hausdorff dimension and conditions for global asymptotic stability