Some new aspects in a classic problem of axial rod compression are presented. For short-term impact the influence of longitudinal waves propagation on the rod instability is investigated. For the long-term compression which considerably exceeds the Euler load, the generalization of the overcritical deformations in the rod from investigated by M.A. Lavrentiev and A.Yu. Ishlinsky in the linear statement to the nonlinear Euler elastics is studied.