The evolution of convergent plate margins includes a series of long-term and successive processes from subduction initiation (SI) through mature oceanic subduction to continental subduction and collision, leading to possible mountain building in different stages. This evolution pathway is one of the most important expressions of plate tectonics, which controls the mass and energy exchange between the Earth's surface and interior and produces the majority of volcanisms and great earthquakes. Based on extensive geological/geophysical observations and geodynamic simulations, we systematically summarize key tectonic and dynamic issues during SI, oceanic subduction, and continental collision and further propose several major unresolved scientific questions and future research directions. (1) For the SI, it generally requires the combination of multiple driving forces and, more importantly, lithospheric weakening. Thereby, it tends to occur at previous suture zones or weak belts. The geological records of SI include magmatic and metamorphic rocks formed in hot environments, as well as tectonic uplift/subsidence and accompanying sedimentary responses in cold environments. According to contrasting geological records, two different types of SI are defined: hot and cold. Hot SI prefers to be driven by a vertical force, whereas cold SI is dominated by horizontal force. (2) For mature oceanic subduction, two typical styles include flat slab subduction beneath an overriding lithosphere and slab stagnation in the mantle transition zone. The favorable conditions of flat subduction include young oceanic slabs, thick oceanic crust, and strong coupling between converging plates. On the other hand, slab stagnation is generally controlled by the resistance from the 660-km discontinuity and the trench retreat. The subducting slab is the major carrier of volatiles (e.g., H2O and CO2) to the Earth's interior. The serpentinite layer beneath the oceanic crust plays a critical role in the transportation of water to the deep mantle, with the thickness and water content of serpentinite as controlling factors. The deep carbon cycling in subduction zone is controlled by multiple processes, including mechanical, metamorphic, dissolution, and partial melting-induced decarbonization. The flux of carbon cycling requires further systematic constraints. (3) For continental subduction, a deeply subducted lithosphere experiences metamorphic phase change and densification, which further results in considerable slab pull. Thereby, when the continental slab is dragged to great depths (e.g., >300 km), continental subduction could be self-sustained. One of the most important continental collisional zones is that of the Himalayas and Tibetan Plateau. Its puzzling geodynamic issues include the mechanisms of multiple slab tearing of the underthrusting Indian plate, the deformation characteristics and material balance of the overriding plateau, and the collision-induced far-field intracontinental orogeny in central Asia. In addition, the coupling between these deep Earth tectonics and surficial processes (e.g., erosion and sedimentation) plays a significant role in shaping the Earth as well as its environment and climate effects. In summary, the research on the dynamics of convergent plate margins requires effective integrations of multiple geological, geophysical, and geochemical observations as well as the coupled thermodynamic and thermomechanical numerical models. They may jointly contribute to the ultimate solution of relevant unresolved scientific issues.