Low-melting-point liquid metal convection is rapidly emerging as a high-performance heat transfer technology in electronics thermal management and energy fields. The advantages of gallium-based and bismuth-based liquid metals, such as low melting point, high thermal conductivity, nonflammability, and nontoxic characteristic, make liquid metals highly attractive for high heat flux density applications at high temperatures. Their convective heat transfer behavior, driving techniques, and application systems are different from those of conventional liquids, such as water, oil, sodium-potassium, and lead-bismuth. Although their significance in both academia and industry has gradually increased, to the best of our knowledge, a systematic description of gallium-based and bismuth-based liquid metal convection and its applications has not been reported thus far. Therefore, in this paper, we present a thorough review of low-melting-point liquid metal convective heat transfer technologies. Specifically, liquid metal fluids and their convection mechanisms are introduced first, followed by the description of typical driving techniques of liquid metals based on electromagnetic, thermal, electrical and magnetic methods. Subsequently, applications of typical liquid metal convection methods in industrial heat transfer and energy fields are presented. Finally, extended liquid metal convection enhancement techniques based on microchannel and jet impingement are interpreted. Both the fundamental mechanisms and recent application research are elaborated, and critical issues are discussed. The scientific and technical challenges, along with future developments in these areas, are also highlighted in the paper.