The doping effect of Sn on the microstructure evolution and dielectric properties was studied in $CaCu_3Ti_{4-x}Sn_xO_{12}$ polycrystals. Samples were produced by a conventional solid-state reaction method. Sintering was carried out at $1115^{circ}C$ for 2-16 h in air. The dielectric constant and loss were examined at room temperature over a frequency range between $10^2$ and $10^6$ Hz. The microstructure was found to evolve into three stages. Addition of $SnO_2$ led to an increase in density and advanced formation of abnormal grains. The formation of coarse grains with a reduced thickness of the boundary brought about an enhanced dielectric constant and a lower dielectric loss below ~1 kHz. EDS data showed the Cu-rich phase along the grain boundary, which should contribute to the improved dielectric constant according to the internal barrier layer capacitor model. After all, $SnO_2$ was an effective dopant to elevate the dielectric characteristics of $CaCu_3Ti_{4-x}Sn_xO_{12}$ polycrystals as a promoter for abnormal grain growth.