The shock waves of supernova remnants (SNRs) are the traditional sources of Galactic cosmic rays, at least up to about 3000 TeV (the "knee" energy in the cosmic-ray spectrum). In the last decade or so, X-ray observations have confirmed in a few SNRs the presence of synchrotron-X-ray-emitting electrons with energies of order 100 TeV. TeV photons from SNRs have been observed with ground-based air Cerenkov telescopes as well, but it is still unclear whether they are due to hadronic processes (inelastic p-p scattering of cosmic-ray protons from thermal gas, with secondary neutral pions decaying to gamma rays), or to leptonic processes (inverse-Compton upscattering of cosmic microwave background photons, or bremsstrahlung). The spatial structure of synchrotron X-rays as observed with the Chandra X-ray Observatory suggests the remarkable possibility that magnetic fields are amplified by orders of magnitude in strong shock waves. The electron spectra inferred from X-rays reach 100 TeV, but at that energy are cutting off steeply, well below the "knee" energy. Are the cutoff processes due only to radiative losses so that ion spectra might continue unsteepened? Can we confirm the presence of energetic ions in SNRs at all? Are typical SNRs capable of supplying the pool of Galactic cosmic rays? Is strong magnetic-field amplification a property of strong astrophysical shocks in general? These major questions require the next generation of observational tools. I shall outline the theoretical and observational framework of particle acceleration to high energies in SNRs, and shall describe how GLAST will advance this field.