Arc plasma, characterized by its high energy density and abundant chemically reactive species, has wide applications in numerous fields including energy and environmental protection, aeronautics and astronautics, advanced materials processing, and national defense. The non-equilibrium feature of arc plasmas is one of the key factors influencing the scope and effectiveness of plasma applications. Taking a direct-current (DC) free-burning arc as a model system for studying non-equilibrium synergistic transports in arc plasmas, this paper provides a comprehensive review of the state-of-the-art of numerical modeling of transport mechanisms and active modulation methods of key parameters of arc plasmas. First, the development of physical-mathematical models and the related research progress of numerical simulations for non-equilibrium transports in DC arc plasmas are summarized, highlighting a shift in research focus from local thermodynamic equilibrium modeling for the arc column region to considering thermal, chemical, or even electrical non-equilibrium effects within the cold wall boundary layer and/or the plasma-cold gas interaction region, and further extending to multi-region-coupled modeling with consideration of the arc column, electrode boundary layer, solid electrode, and external circuit. Consequently, based on systematic discussions of the synergistic mass-momentum-energy transport mechanism and its influences on the characteristics of arc plasma systems, a summary of current active regulation methods for the key parameters of arc plasmas is presented, e.g., with the aid of geometrical design of plasma generators, variation of external electromagnetic field configurations, and adjustment of operating parameters. Finally, the key scientific issues for promoting fundamental research and industrial applications of arc plasmas are discussed briefly, e.g., development of multi-region, multi-phase coupled numerical models; high-efficiency numerical methods with high spatiotemporal resolutions; high-efficiency chemical reaction pathway screening methods; reliable databases for chemical reaction kinetics and related thermodynamic and transport properties of plasmas; and even creation of a new research paradigm for analyzing complicated mass-energy synergistic transport mechanisms and for developing novel arc plasma sources with desired parameters facing various application demands.