Over the past thirty years, a consensus has evolved that cosmic rays below about 3000 TeV are produced by strong shock waves in young supernova remnants (SNRs), based largely on the observed nonthermal X-rays and gamma-rays from SNRs. I shall describe theoretical modeling and observations indicating that the diffusive shock acceleration process can produce particles of the required energies. Obeservations imply that magnetic fields are amplified by large factors in very strong collisionless shocks. Evidence is mounting that energetic ions as well as electrons must be present in SNRs, but an incontrovertible demonstration is still lacking.