"Acceleration of particles at high gradient is an intense research topic. Without new acceleration methods allowing for future accelerators to be of reasonable size and cost, accelerator-based high-energy physics may reach an end. Plasmas can sustain very large longitudinal electric fields that can travel at velocities close to the speed of light. Large energy gains where demonstrated for electrons in plasma-based accelerators driven by intense laser pulses or short electron bunches. Research on the beam-driven, plasma wakefield accelerator (PWFA) scheme is experiencing an increasing interest. This is motivated by the 42GeV energy gain in 85 cm of plasma (50GeV/m accelerating gradient) demonstrated at SLAC [Blumenfeld, Nature 445, 741 (2007)]. Current PWFA experiments focus on two aspects: first, on the production of high quality accelerated bunches (narrow energy spread, low emittance, etc.) for example at SLAC FACET; second, on a new scheme based on the transverse self-modulation of long particle bunches in dense plasmas to resonantly drive wakefields [Kumar, Phys. Rev. Lett. 104, 255003 (2010)]. After briefly introducing the PWFA and the recent results I will focus on the second aspect and begin by introducing the physics of the self-modulation instability (SMI). I will discuss initial experiments at the Brookhaven National Laboratory Accelerator Test Facility that have shown that long electron bunches do indeed drive multiple period wakefields. They also suggest that the seeding of the self-modulation instability is possible. After presenting these results, I will describe experiments that we are planning at SLAC-FACET to study the physics of the SMI of electron and positron bunches. This experimental program is known as E209. Then I will describe the AWAKE project that was recently approved at CERN. AWAKE will use the long SPS, 400GeV bunch with 3x10^11 protons and a 10m-long plasma to address the issues related to driving GeV/m accelerating gradients over large distances to accelerate electrons to high energies in a single plasma section. This experiment will operate at lower gradient than other plasma-based accelerators, but aims at avoiding the staged-acceleration necessary when using drivers (laser pulse or particle bunch) carrying small amounts of energy. The SPS bunch carries ~20kJ, while LHC bunches carry more than 100kJ, more than necessary to produce an ILC-like electron bunch (2x10^10 e -500GeV or ~1.6kJ)! Operating at lower plasma density and therefore with a larger accelerating structure also eases the injection process and the beams generation and alignment tolerances. The purpose of the presentation is to give an overview of the experimental programs we are developing while explaining the basic concepts."