High-energy physics is at the brink of very exciting discoveries. These are enabled by forefront accelerator technology. A linear lepton collider is widely recognized as the natural complement to a circular hadron collider such as the LHC. However, future linear colliders at the energy frontier using the current radio frequency accelerator technology maybe out of reach because of their large size and prohibitive cost. Advanced acceleration techniques must be invented and demonstrated to reduce the size and cost of the accelerator sections of such a collider. Among these advanced techniques, plasma based-acceleration has shown rapid and remarkable progress over the last few years. A scheme known as the plasma wakefield accelerator (PWFA) uses a short, tightly focused, high-charge, relativistic particle bunch to drive large amplitude wakefields in a high density plasma. These wakefields have large longitudinal components traveling near the speed of light that can therefore accelerate a trailing bunch to very high energies over a short distance. Experiments using the 28-42GeV electron and positron bunches form the SLAC linac have demonstrated, among many other results, that the energy of trailing electrons can be increased by 42GeV in less than one meter of plasma (I. Blumenfeld et al., Nature 445, 741 (2007)). Experiments with low energy beams (~60MeV) are a test bed for the physics of the PWFA and for techniques that will be implemented to improve experiments at high energy. Pioneering experiments aiming at extracting the large energy of LHC type proton bunches are also proposed to explore the possibility of producing TeV-class electron bunches in a single PWFA stage. The PWFA scheme will be described, and a brief history presented. A summary of the PWFA results recently obtained with low and high-energy electron and positron bunches will be given. Plans for PWFA experiments at CERN that we are proposing will be outlined. Challenges that the PWFA will have to answer to become the new accelerating technology for a future electron/positron collider will be whispered.