AbstractMemristors are non-volatile nano-resistors which resistance can be tuned by applied currents or voltages and set to a large number of levels. Thanks to these properties, memristors are ideal building blocks for a number of applications such as multilevel non-volatile memories and artificial nano-synapses, which are the focus of this work. A key point towards the development of large scale memristive neuromorphic hardware is to build these neural networks with a memristor technology compatible with the best candidates for the future mainstream non-volatile memories. Here we show the first experimental achievement of a multilevel memristor compatible with spin-torque magnetic random access memories. The resistive switching in our spin-torque memristor is linked to the displacement of a magnetic domain wall by spin-torques in a perpendicularly magnetized magnetic tunnel junction. We demonstrate that our magnetic synapse has a large number of intermediate resistance states, sufficient for neural computation. Moreover, we show that engineering the device geometry allows leveraging the most efficient spin torque to displace the magnetic domain wall at low current densities and thus to minimize the energy cost of our memristor. Our results pave the way for spin-torque based analog magnetic neural computation.