This paper presents a mathematical model for the coupled electromagnetic and thermal analysis of the single-phase insulated high-current busducts of circular cross-section geometry and of gas-insulated transmission lines (GIL). The mathematical model, accompanied by a numerical solution procedure, features an exact current distribution in phase conductors and shields of the busduct or GIL system, accounting for the skin and proximity effects, and including the complete electromagnetic coupling between phase conductors and shields. The current distribution is based on the conductor filament method in combination with the mesh-current method. The mathematical model further couples the analysis of current distribution with the computation of (Joule) power losses and subsequent temperature increase in the high-current busducts or GIL systems, accounting for the material properties (electrical conductivity, thermal emission and convection coefficients) as well as for the surrounding ambient properties (ambient temperature, wind and solar radiation influences).