Abstract: Unlike grid-following converters, the grid-forming converter is operated as a synchronous voltage source, which can effectively improve the stability of power-electronics-dominated power systems, and hence, has attracted much research attention in recent years. As a basis of the large-scale implementation of grid-forming converters, there should be an in-depth understanding of its stability under different grid strengths as well as different grid disturbances. This article performs a comprehensive overview of the stability studies of grid-forming converters from the perspective of existence of equilibrium points, small-signal stability, and transient stability. On top of this, stabilization control solutions of grid-forming converters are also summarized. In the end, perspectives on the prospects and challenges of stability analysis of grid-forming converters are shared.