A quantitative description of the transport properties, diffusivity, viscosity, electrical, and thermal conductivity, of silicate melts is essential for understanding melting-related petrologic and geodynamic processes. We here provide a systematic overview on the current knowledge of these properties from experiments and molecular dynamics simulations, their dependence on pressure, temperature, and composition, atomistic processes underlying them, and physical models to describe their variations. We further establish phenomenological and physical links between diffusivity, viscosity, and electrical conductivity that are based on structural rearrangement in the melt. Neutral molecules and network-modifying cations with low electric field strength display intrinsic diffusivity, which is controlled by the intrinsic properties (size and valence) of the species. By contrast, oxygen and network formers with high field strength show extrinsic diffusivity, which is more sensitive to extrinsic parameters including temperature (T), pressure (P), and melt composition (X). Similar T-P-X dependence of diffusivity and electrical conductivity and their quantitative relation reveal the role of intrinsically diffusing species in electrical transport, while viscosity is tied to the extrinsically diffusing species in a similar way. However, the differences in the structural role and mobility of various atomic species diminish with increasing temperature and/or pressure: all transport processes are increasingly coupled, eventually converging to a uniform rate and mechanism. Accurate comprehension of interatomic interactions and melt structure is vital to fully accounting for the compositional dependence of transport properties, and simple polymerization parameters such as nonbridging oxygen per tetrahedrally coordinated cation are inadequate.