Plasmonic effects have been proposed as a solution to overcome the limited light absorption in thin-film photovoltaic devices, and various types of plasmonic solar cells have been developed. This review provides a comprehensive overview of the state-of-the-art progress on the design and fabrication of plasmonic solar cells and their enhancement mechanism. The working principle is first addressed in terms of the combined effects of plasmon decay, scattering, near-field enhancement, and plasmonic energy transfer, including direct hot electron transfer and resonant energy transfer. Then, we summarize recent developments for various types of plasmonic solar cells based on silicon, dye-sensitized, organic photovoltaic, and other types of solar cells, including quantum dot and perovskite variants. We also address several issues regarding the limitations of plasmonic nanostructures, including their electrical, chemical, and physical stability, charge recombination, narrowband absorption, and high cost. Next, we propose a few potentially useful approaches that can improve the performance of plasmonic cells, such as the inclusion of graphene plasmonics, plasmon-upconversion coupling, and coupling between fluorescence resonance energy transfer and plasmon resonance energy transfer. This review is concluded with remarks on future prospects for plasmonic solar cell use.