Abstract: The teaching issues are addressed for the stability and the instability of equilibriums in a potential field. Firstly, relevant contents are collected from classic works by masters such as Lagrange, Lyapunov, and Chetaev and a few landmark papers and treats in order to clarify the fundamental facts regarding the stability and the instability of equilibriums. Then the teaching treatments on the topic are summarized from the domestic textbooks of theoretical mechanics, and the overview indicates the weakness in the instability teaching. Rectilinearly moving particle in a potential filed is treated as an example of a single-single-degree-of-freedom system to demonstrate intuitively the relationship between the potential energy extremum and the equilibrium stability. Finally, it is suggested that the essential teaching contents should include the stability conditions of multi-degree-of-freedom systems and the instability conditions of single-degree-of-freedom systems.