Grid-connected converters exposed to weak grid conditions and severe fault events are at risk of losing synchronism with the external grid and neighboring converters. This predicament has led to a growing interest in analyzing the synchronization mechanism and developing models and tools for predicting the transient stability of grid-connected converters. This paper presents a thorough review of the developed methods that describe the phenomena of synchronization instability of grid-connected converters under severe symmetrical grid faults. These methods are compared where the advantages and disadvantages of each method are carefully mapped. The analytical derivations and a detailed simulation model are verified through experimental tests of three case studies. Steady-state and quasi-static analysis can determine whether a given fault condition results in a stable or unstable operating point. However, without considering the dynamics of the synchronization unit, transient stability cannot be guaranteed. By comparing the synchronization unit to a synchronous machine, the damping of the phase-locked loop is identified. For accurate stability assessment, either nonlinear phase portraits or time-domain simulations must be performed. Until this point, no direct stability assessment method is available which consider the damping effect of the synchronization unit. Therefore, additional work is needed on this field in future research.