Broadband low-coherence light is expected to overcome the laser-plasma instability in inertial confinement fusion. However, achieving an efficient third-harmonic generation of broadband low-coherence light remains a significant technical problem in engineering applications. Currently, limited research has been done on the third-harmonic generation of broadband low-coherence light. In this study, the characteristics of the broadband low-coherence light are introduced, along with the frequency conversion processes involving second-harmonic and sum frequency generation. Subsequently, the fundamental challenges for an efficient third-harmonic generation with broadband low-coherence light are presented and an overview of the developmental history, principles, and limitations of various broadband third-harmonic generation schemes is provided. In experiments on the frequency conversion of low-coherence light, the acceptance bandwidth and nonlinear coefficients of commonly used nonlinear frequency-conversion crystals are compared for reference. Finally, some suggestions for potential breakthrough directions are proposed for an efficient third-harmonic generation of broadband low-coherence light.