The mechanism of flame spread over thermally thick solid fuels in an opposing flow is explored. The pathways of forward heat transfer are computed with mathematical models of increasing complexity in all regimes. New formulas are developed that reproduce numerical results for the dominant forward heat transfer pathways in the thermal and chemical kinetic regimes and compare well with experimental data. The heat transfer structure at the leading edge of the flame computed by the comprehensive model agrees quite well with available experimental and numerical results. In the thermal regime, forward heat transfer through the gas is dominant, while in the chemical kinetic and microgravity regimes, forward heat transfer through the solid may assume a significant role. In the latter regime the contribution of forward radiation heat transfer is found to be negligible, although gas-phase radiation has a profound effect on the flame structure. The sensitivity of the dimensionless contribution of forward heat transfer through the solid fuel to that through the gas, R-sfc, on the location of the flame leading edge reconciles seemingly contradictory results found in the literature. A semi-empirical predictive formula for R-sfc is developed supported by numerical and experimental results. The result, expressed as: R-sfc=c(sfc)(Q(sfc)/Q(gfc))=c(sfc)(lambda(s) /lambda(g))(1/F(L-s/max(L-s,L-g))(0.5), limiting forms of this quantity are obtained for the thermal regime and the chemical kinetic blow-off limit; R-sfc,R-thermal = c(sfc)(lambda(s)/lambda(g)F)(2) and R-sfc,R-blowoff= c(sfc)(lambda(s)/lambda(g))(1/F). Values of both limiting forms can be determined from known parameters of the problem.