The status and evolution of the electron beam-driven Plasma Wakefield Acceleration scheme is described. In particular, the effects of the radial electric field of the wake on the drive beam such as multiple envelope oscillations, hosing instability and emission of betatron radiation are described. Using ultra-short electron bunches, high-density plasmas can be produced by field ionization by the electric field of the bunch itself. Wakes excited in such plasmas have accelerated electrons in the back of the drive beam to greater that 4 GeV in just 10 cm in experiments carried out at the Stanford Linear Accelerator Centre.