The paper contains a short survey of the papers on the static and dynamic longitudinal compression of a thin rod initiated by Morozov and and carried out in 2009–2016 with his direct participation. We consider linear and nonlinear problems related to the propagation of longitudinal waves in a rod and the transverse vibrations generated by these waves; parametric resonances; beating due to energy exchange between longitudinal and transverse vibrations; the rod shape evolution as the load exceeds the Euler critical value; the possibility of buckling of the rod rectilinear shape under a load less than the Euler load; and the rod dynamics at the initial stage of motion. The prospects of further investigations related to the complication of the models are considered, in particular, the problem of longitudinal impact by a body on a rod and the transverse vibrations generated by it.