This paper reviews the concepts, recent progress and current challenges for realizing the tremendous electric fields in relativistic plasma waves for applications ranging from tabletop particle accelerators to high-energy physics. Experiments in the 1990s on laser-driven plasma wakefield accelerators at several laboratories around the world demonstrated the potential for plasma wakefields to accelerate intense bunches of self-trapped particles at rates as high as 100 GeV m−1 in millimetre-scale gas jets. These early experiments have been followed in the current decade by experiments that are advancing on several fronts—increasing the accelerated charge (to the several nanocoulomb level), producing higher transverse beam quality (to the mm mrad normalized emittance level) and accessing new physics regimes at higher laser power. Several groups are engaged in pursuing two key challenges for laser wakefield accelerators—producing beams with small energy spread and extending the interaction length from millimetres to centimetres and beyond. Major breakthroughs on both fronts have occurred in the past year. In parallel with the progress in laser-driven wakefields, particle-beam driven wakefield accelerators are making large strides. A series of experiments using the 30 GeV beam of the Stanford Linear Accelerator has demonstrated high-gradient acceleration of electrons and positrons in metre-scale plasmas as well as key scaling laws for a 'plasma afterburner', a concept for doubling the energy of a high-energy collider in a few tens of metres of plasma. In addition to wakefield acceleration, these and other experiments have demonstrated the rich physics bounty to be reaped from relativistic beam–plasma interactions. This includes plasma lenses capable of focusing particle beams to the highest energy density ever produced, collective radiation mechanisms capable of generating high-brightness x-ray beams, collective refraction of particles at a plasma interface and acceleration of intense proton beams from laser-irradiated foils.