With the development of power electronic technology and requirement for clean energy, the traditional power systems which are dominated by synchronous generators are gradually changing into the power-electronic-based power systems with diversified power electronic equipment. The power systems are facing a great revolution in their primary equipment, and this has not happened in the past one hundred years. In recent years, with great increasing penetration of power electronic devices into power grids, the large-scale blackouts caused by power electronic devices have been reported, which seriously threatens the safe and stable operation of power systems. Under the above background, in this paper we first introduce several methods of analyzing the traditional power system transient stability from the equal area criterion for the single machine infinite bus system to several Lyapunov function based direct methods for multi-machine systems. Then we introduce some of our recent work on the nonlinear modeling and analysis of a key component of power-electronic-based power systems, voltage source converter (VSC), and propose a multiple machine system model including power electronic equipment and traditional synchronous machines. Finally, we illustrate the transient characteristics of the power electronic devices, and summarize the basic problems and challenges for the transient stability of power-electronic-based power systems. We hope that these basic problems in power-electronic-based power system dynamics including nonlinearity, multi-time-scale, and complexity could arouse the general interest of researchers in the fields of complex systems and statistical mechanics.