Silicon-based optoelectronic integration technology fully utilizes the advantages of low power consumption, fast response, and low signal crosstalk of optical interconnects. This technology is one of the significant solutions to break through the bottleneck of current development in electrical interconnect integrated circuits. Among them, silicon-based optical waveguide amplifiers containing rare-earth ions play a key role in achieving high-power and low-loss on-chip optical interconnects as the core device. This article reviews the research progress of silicon-based optical waveguide amplifiers and discusses three main research directions: silicon-based hybrid integrated III-V semiconductor optical waveguide amplifiers, silicon-based nonlinear Raman optical amplifiers, and silicon-based rare-earth ion-containing optical waveguide amplifiers. The research focuses specifically on silicon-based rare-earth ion optical waveguide amplifiers, exploring in detail the development history of their gain materials, waveguide structures, and integration platforms. The advantages and challenges of this device in providing optical gain and promoting single-chip silicon optical integration are also elaborated. We point out that rare-earth ion waveguide amplifiers are the key to promoting the practical application of silicon-based optoelectronic integration, and their development will provide important technical support for high-tech fields such as optical computing and data centers in the post-Moore's era.