We report GeSn p-i-n resonant-cavity-enhanced photodetectors (RCEPDs) grown on silicon-on-insulator substrates. A vertical cavity, composed of a buried oxide as the bottom reflector and a deposited SiO2 layer on the top surface as the top reflector, is created for the GeSn p-i-n structure to enhance the light-matter interaction. The responsivity experiments demonstrate that the photodetection range is extended to 1820 nm, completely covering all the telecommunication bands, because of the introduction of 2.5% Sn in the photon-absorbing layer. In addition, the responsivity is significantly enhanced by the resonant cavity effects, and a responsivity of 0.376 A/W in the telecommunication C-band is achieved that is significantly higher than that of conventional GeSn-based PDs. These results demonstrate the feasibility of CMOS-compatible, high-responsivity GeSn-based PDs for shortwave infrared applications.