1 Development of the theory of motion for systems with Coulomb friction.- 1.1 Coulomb’s law of friction.- 1.2 Main peculiarities of systems with Coulomb friction and the specific problems of the theory of motion.- 1.3 Various interpretations of Painlevé’s paradoxes.- 1.4 Principles of the general theory of systems with Coulomb friction.- 1.5 Laws of Coulomb friction and the theory of frictional selfexcited oscillations.- 2 Systems with a single degree of freedom and a single frictional pair.- 2.1 Lagrange’s equations with a removed contact constraint.- 2.2 Kinematic expression for slip with rolling.- 2.3 Equation for the constraint force and Painlevé’s paradoxes.- 2.4 Immovable contact and transition to slipping.- 2.5 Self-braking and the angle of stagnation.- 3 Accounting for dry friction in mechanisms. Examples of single-degree-of-freedom systems with a single frictional pair.- 3.1 Two simple examples.- 3.2 The Painlevé-Klein extended scheme.- 3.3 Stacker.- 3.4 Epicyclic mechanism with cylindric teeth of the involute gearing.- 3.5 Gear transmission with immovable rotation axes.- 3.6 Crank mechanism.- 3.7 Link mechanism of a planing machine.- 4 Systems with many degrees of freedom and a single frictional pair. Solving Painlevé’s paradoxes.- 4.1 Lagrange’s equations with a removed constraint.- 4.2 Equation for the constraint force, differential equation of motion and the criterion of paradoxes.- 4.3 Determination of the true motion.- 4.4 True motions in the Painlevé-Klein problem in paradoxical situations.- 4.5 Elliptic pendulum.- 4.6 The Zhukovsky-Froude pendulum.- 4.7 A condition of instability for the stationary regime of metal cutting.- 5 Systems with several frictional pairs. Painlevé’s law of friction. Equations for the perturbed motion takingaccount of contact compliance.- 5.1 Equations for systems with Coulomb friction.- 5.2 Mathematical description of the Painlevé law of friction.- 5.3 Forces of friction in the Painlevé-Klein problem.- 5.4 The contact compliance and equations of perturbed trajectories.- 5.5 Painlevé’s scheme with two frictional pairs.- 5.6 Sliders of metal-cutting machine tools.- 5.7 Concluding remarks about Painlevé’s paradoxes.- 6 Experimental investigations into the force of friction under self-excited oscillations.- 6.1 Experimental setups.- 6.2 Determining the forces by means of an oscillogram.- 6.3 Change in the force of friction under break-down of the maximum friction in the case of a change in the velocity of motion.- 6.4 Dependence of the friction force on the rate of tangential loading.- 6.5 Plausibility of the dependence F+(f).- 6.6 Characteristic of the force of sliding friction.- 7 Force and small displacement in the contact.- 7.1 Components of the small displacement.- 7.2 Remarks on friction between steel and polyamide.- 7.3 Conclusions.- 8 Frictional self-excited oscillations.- 8.1 Self-excited oscillations due to hard excitation.- 8.2 Self-excited oscillations under both hard and soft excitations.- 8.3 Accuracy of the displacement.- References.