New grid devices based on power electronics technologies are increasingly emerging and introduce two new types of stability issues into power systems, which are different from traditional power system stability phenomena and not well understood from a system perspective. This paper intends to provide the state of the art on this topic with a thorough and detailed review of the converter-driven stability issues in partial or all power electronics-based grids. The underlying and fundamental mechanisms of the converter-driven stability issues are uncovered through different types of root causes, including converter controls, grid strength, loads, and converter operating points. Furthermore, a six-inverter two-area meshed system is constructed as a representative test case to demonstrate these unstable phenomena. Finally, the challenges to cope with the converter-driven stability issues in future power electronics-based grids are identified to elucidate new research trends.