Cosmic rays, the product of natural extraterrestrial particle accelerators far more powerful than the LHC, were first detected a century ago. A "standard model" of cosmic-ray acceleration in supernova remnants has begun to emerge, but a number of questions still require satisfactory answers. The maximum particle energy attainable via the most favored mechanism, diffusive shock acceleration, is limited by the amplitude of magnetic-field turbulence in the unshocked interstellar or circumstellar medium, but cosmic rays are observed at high enough energies that some magnetic-field amplification is required. By what mechanisms might this amplification occur, and can it operate to a great enough extent to account for those cosmic rays thought to be of Galactic origin? A number of proposed solutions involve instabilities arising from interactions between cosmic rays and the upstream plasma, whose evolution becomes highly nonlinear. A related question explored is whether the presence of accelerated particles in the shock vicinity has any microscopic effect on the instabilities governing the shock itself. Particle-in-cell kinetic simulations allow us to investigate the growth and saturation of these instabilities at the (astrophysically) microscopic scale, providing valuable insights and important considerations for self-consistent macroscopic models of particle acceleration.