There are many different research routes that are investigating how to help with treating and hopefully eventually curing spinal cord injury.
One very exciting avenue is the use of implants, either electronic devices or biological tissue, which can bypass or repair the area of the injury. This would mean that the signals from the brain that could not travel to the limbs would be restored and the paralysed limbs would then function as normal. Many labs are researching these aspects.
Other research efforts concentrate on potential for rehabilitation of existing connections. What this research tries to achieve is to strenghten the remaining nerves, if any, at the place of the injury. By recording signal from the brain regarding movement (“think about moving your leg”) and then decoding this information and making the appropriate limb move (through electrical stimulation or a robotic device) it might be possible to improve any little connections between the brain and the limb, restoring some movement and sensation.
Overall, I think the next two decades will be very exciting for research on spinal cord injury, as many different research disciplines are coming together for potential solutions.