[Objective] With the increasing penetration of renewable energy， the power system is accelerating its transformation towards being dominated by power electronic equipment. As the core interface for renewable energy grid connection， the transient stability of converter-connected systems has become a critical issue restricting the safe operation of new-type power systems. Grid-following （GFL） and grid-forming （GFM） converters represent two mainstream grid-connected converter paradigms， with fundamental differences in synchronization mechanisms， dynamic characteristics and instability mechanisms. A systematic analysis of their transient stability mechanisms， a comparison of evaluation methods， and a summary of enhancement strategies are essential for ensuring the safe operation of converter grid-connected systems. [Methods] Based on the differences in synchronization mechanisms and control structures， this paper delineates the core differences in transient behaviors between phase-locked loop （PLL） synchronization control for grid-following converters and power synchronization control （PSC） for grid-forming converters. It elucidates the influence mechanisms of PLL dynamics， equivalent inertia and damping characteristics， and current limiting on transient stability. On this basis， three categories of transient stability analysis methods are systematically summarized： methods based on solving differential-algebraic equations， energy function methods， and artificial intelligence methods. Their applicable scopes， advantages and limitations are also concluded. [Results] The research shows that GFL converter transient stability is highly sensitive to grid strength and disturbance intensity， and is prone to synchronization failure due to PLL out-of-step under weak grids or large disturbances； GFM converters can actively provide voltage and frequency references， but exhibit complex nonlinear dynamic characteristics due to control coupling in multi-machine interconnection scenarios， and current limiting may easily induce unexpected equilibrium points and mode-switching problems. In general， current research still suffers from three major shortcomings： insufficient understanding of the mechanism of transient synchronization instability under strongly nonlinear control， the lack of transient stability criteria applicable to power-electronics-dominated systems， and the absence of a global decentralized coordinated control framework for multi-source. [Conclusions] This paper systematically summarizes the research progress in transient stability of grid-connected systems containing GFL/GFM converters， and points out key directions for future research： characterization of transient stability mechanisms under different operating modes， construction of coupling mechanisms and transient stability criteria for multi-machine systems， decentralized coordinated control strategies for stability enhancement with weak parameter dependence， and assessment and control methods combining physical mechanisms with data-driven approaches. These conclusions can provide methodological references and a basis for design， while deepening the understanding of transient stability in power-electronics-dominated systems and supporting the safe operation of new-type power systems.