Abstract: Power system shows new characteristics such as weak grid strength, low inertia and damping level, and weak anti-disturbance ability due to the increasing proportion of renewable energy. Grid-forming converters which offers inertia and damping supports to renewable energy power system become prominent, making them the most promising type of converters. Converters play an important role in transmitting electricity to the grid safely, stably, and efficiently as the interface between renewable energy and the power system. The stability of converters greatly affects power system with high proportion of renewable energy. In this article, we first focus on the modeling strategy and disturbance analysis method of grid-forming converters and summarize the stability control strategies of grid-forming converters. Then, on the basis of the relations between controller and stability, we reveal the dynamic and transient characteristics of grid-forming converters in three aspects: inertia dynamics, fault current, and broadband oscillation. Furthermore, we clarify the key issues of stable control for grid-forming converters under large-scale power grid, put forward to the future research directions of stability for grid-forming converters, and provide a reference for further research and application of grid-forming converters.