A magnetic Bloch line (BL) is a nanoscale topological defect with a nontrivial topological charge. Herein, we propose a novel memory that utilizes current-driven BL motion. We demonstrate that it is possible to write and shift BLs using a spin-transfer-torque scheme. The writing efficiency and shifting velocity of the BL-based racetrack memory are better than or comparable to those of domain wall (DW)-based racetrack memory. Moreover, the current-driven BL motion is more robust against defects than the DW motion, suggesting that BL-motion-based memory can serve as a next-generation memory beyond the DW-motion-based racetrack memory.