Abstract: Convergent plate margins occur when two adjoining tectonic plates come together to form either a subduction zone, where at least one of the converging plates is oceanic and plunges beneath the other into the mantle, or a collision zone, where two continents or a continent and a magmatic arc collide. Convergent plate margins are arguably the most complicated and dynamic plate boundaries on Earth and have been the subject of many investigations and discussions since the advent of plate tectonic theory. This paper provides a historical background and a review of the development of geological and geodynamic theories on convergent plate margins. Furthermore, it discusses some of the recent advances that have been made in the fields of structural geology, geophysics and geodynamics, which are fundamental to our understanding of this phenomenon. These include: (1) the finding that plates and plate boundaries move at comparable velocities across the globe; (2) the emerging consensus that subducted slabs are between two to three orders of magnitude stronger than the ambient upper mantle; (3) the importance of lateral slab edges, slab tearing and toroidal mantle flow patterns for the evolution of subduction zones; and (4) clear evidence from mantle tomography that slabs can penetrate into the lower mantle. Still, many first-order problems regarding the geodynamic processes that operate at convergent margins remain to be solved. These include subduction zone initiation and the time of inception of plate tectonics, and with it convergent plate margins, on Earth.