This chapter presents the classic results by Euler on nonlinear static deformation of the axially compressed rod, the results of the work by M.A. Lavrentiev and A.Yu. Ishlinsky on rod buckling under dynamic axial compression and the recent results mainly related to the study of interaction of longitudinal and transverse vibrations. The chapter introduces readers to the basic research methods which are the D’Alembert and Fourier methods, the methods of studying parametric resonances, the asymptotic method of two-scale expansion.