Studies of charged-particle acceleration processes remain one of the most important areas of research in laboratory, space and astrophysical plasmas. In this paper, we present the underlying physics and the present status of high gradient and high energy plasma accelerators. We will focus on the acceleration of charged particles to relativistic energies by plasma waves that are created by intense laser and particle beams. The generation of relativistic plasma waves by intense lasers or electron beams in plasmas is important in the quest for producing ultra-high acceleration gradients for accelerators. With the development of compact short pulse high brightness lasers and electron positron beams, new areas of studies for laser/particle beam-matter interactions is opening up. A number of methods are being pursued vigorously to achieve ultra-high acceleration gradients. These include the plasma beat wave accelerator mechanism, which uses conventional long pulse (∼100 ps) modest intensity lasers (I ∼ 1014–1016 W cm−2), the laser wakefield accelerator (LWFA), which uses the new breed of compact high brightness lasers (<1 ps) and intensities >1018 W cm−2, the self-modulated LWFA concept, which combines elements of stimulated Raman forward scattering, and electron acceleration by nonlinear plasma waves excited by relativistic electron and positron bunches. In the ultra-high intensity regime, laser/particle beam–plasma interactions are highly nonlinear and relativistic, leading to new phenomena such as the plasma wakefield excitation for particle acceleration, relativistic self-focusing and guiding of laser beams, high-harmonic generation, acceleration of electrons, positrons, protons and photons. Fields greater than 1 GV cm−1 have been generated with particles being accelerated to 200 MeV over a distance of millimetre. Plasma wakefields driven by positron beams at the Stanford Linear Accelerator Center facility have accelerated the tail of the positron beam. In the near future, laser plasma accelerators will be producing GeV particles.