Domain walls, nanoscale transition regions separating oppositely oriented ferromagnetic domains, have significant promise for use in spintronic devices for data storage and memristive applications. The state of these devices is related to the wall position and thus rapid operation will require a controllable onset of domain wall motion and high speed wall displacement. These processes are traditionally driven by spin transfer torque due to lateral injection of spin polarized current through a ferromagnetic nanostrip. However, this geometry is often hampered by low maximum wall velocities and/or a need for prohibitively high current densities. Here, using time-resolved magnetotransport measurements, we show that vertical injection of spin currents through a magnetic tunnel junction can drive domain walls over hundreds of nanometers at ~500 m/s using current densities on the order of 6 MA/cm(2). Moreover, these measurements provide information about the stochastic and deterministic aspects of current driven domain wall mediated switching.