Spinal cord injury (SCI) is a destructive condition that results in lasting neurological damage resulting in disruption of the connection between the central nervous system and the rest of the body. Currently, there are several approaches in the treatment of a damaged spinal cord; however, none of the methods allow the patient to return to the original full-featured state of life before the injury. Cell transplantation therapies show great potential in the treatment of damaged spinal cords. The most examined type of cells used in SCI research are mesenchymal stromal cells (MSCs). These cells are at the center of interest of scientists because of their unique properties. MSCs regenerate the injured tissue in two ways: (i) they are able to differentiate into some types of cells and so can replace the cells of injured tissue and (ii) they regenerate tissue through their powerful known paracrine effect. This review presents information about SCI and the treatments usually used, aiming at cell therapy using MSCs and their products, among which active biomolecules and extracellular vesicles predominate.