Plasma wakefield acceleration is a novel approach to accelerating charged particles with gradients of the order of GeV/m, paving the way for more compact and cost-effective next-generation particle accelerators. AWAKE, a plasma wakefield acceleration experiment at CERN, is the only one worldwide that uses proton bunches to excite wakefields. This is because the available proton bunches enable the excitation of GV/m fields over distances of hundreds of meters within a single plasma. However, because these proton bunches are initially much longer than the plasma wavelength, they must first undergo self-modulation. This colloquium will summarize the physics concepts underlying the AWAKE experiment and highlight key experimental results, including the observation and stabilization of proton bunch self-modulation, electron acceleration in both uniform plasmas and plasmas with density gradients and steps, as well as the use of a scalable discharge plasma source. Additionally, AWAKE's plans to advance this acceleration scheme for applications in particle physics, as well as design studies for a plasma wakefield-based lepton collider at the 10 TeV scale, will also be presented.