<p>In this paper, we develop the nonlinear dynamics of one-dimensional deformable systems (strings, ropes, rods, beams) with moving boundaries. We eliminate fundamental restrictions of classical linear models that ignore the mutual influence of longitudinal and transverse oscillations, energy exchange across a moving boundary, geometric nonlinearity, and dissipative factors. Important special cases are studied in detail: longitudinal-transverse oscillations of a variable-length string under various types of boundary conditions, the longitudinal dynamics of a rope, and the transverse oscillations of a beam with a moving spring-loaded support carrying an attached mass. For the numerical solution of the nonlinear problems, an original finite-difference method is proposed, whose fundamental feature is the use of an adaptive spatiotemporal grid.</p>