The paper discusses the amplification of magnetic fields and the diffusive shock acceleration of cosmic rays (CR) by supernova remnants (SNR). It explores how CR can generate magnetic fluctuations upstream of a shock, leading to magnetic field amplification. Non-linear simulations show that the magnetic field can be amplified by orders of magnitude from its initial seed value. This has important implications for the maximum energy attainable by CR in SNR. The standard theory of diffusive shock acceleration (DSA) is challenged by the need to reach high energies, and the observed CR energy spectrum suggests that DSA at the outer shocks of SNR is responsible for galactic CR acceleration. However, the standard theory must be stretched to its limits to reach these energies, and the observational support is not unambiguous. There is speculation that acceleration by SNR may continue to energies of $10^{17}-10^{18}$ eV, although the CR at the higher end may be heavier ions. The basic theoretical limit to CR energy arises from the CR Larmor radius, which must be smaller than the acceleration region dimensions. For a proton with energy $10^{15}$ eV in a typical interstellar magnetic field of $3 \mu G$, the Larmor radius is 0.36 pc, which is small compared to the radius of a suitable SNR. However, if the magnetic field is amplified, this limit can be overcome. The paper investigates the possibility that the magnetic field in the acceleration region is amplified by the action of CR. Radio observations suggest that fields as large as $1 mG$ are present in some young SNR. The interaction of CR with the upstream plasma generates strong turbulence, which could wind up a large 'frozen-in' magnetic field. A non-linear theory of CR-driven turbulence is needed. The paper shows that for shocks with high Alfvén Mach number, the process by which streaming CR excite MHD turbulence is different from that usually supposed. This allows for a non-linear simulation with realistic parameters, showing that strong magnetic field amplification can be expected in young SNR. The paper also discusses the linear theory of magnetic field generation and the behavior of waves in different regimes. It shows that the maximum growth rate of waves occurs in a specific regime, and that the growth rate increases with the square root of the CR current in this regime. The paper concludes that the maximum attainable CR energy is limited by the condition that the acceleration time must be less than the age of the SNR. This is determined by the ratio of the Larmor radius to the SNR radius. The paper also discusses the importance of magnetic field amplification in overcoming this limit.The paper discusses the amplification of magnetic fields and the diffusive shock acceleration of cosmic rays (CR) by supernova remnants (SNR). It explores how CR can generate magnetic fluctuations upstream of a shock, leading to magnetic field amplification. Non-linear simulations show that the magnetic field can be amplified by orders of magnitude from its initial seed value. This has important implications for the maximum energy attainable by CR in SNR. The standard theory of diffusive shock acceleration (DSA) is challenged by the need to reach high energies, and the observed CR energy spectrum suggests that DSA at the outer shocks of SNR is responsible for galactic CR acceleration. However, the standard theory must be stretched to its limits to reach these energies, and the observational support is not unambiguous. There is speculation that acceleration by SNR may continue to energies of $10^{17}-10^{18}$ eV, although the CR at the higher end may be heavier ions. The basic theoretical limit to CR energy arises from the CR Larmor radius, which must be smaller than the acceleration region dimensions. For a proton with energy $10^{15}$ eV in a typical interstellar magnetic field of $3 \mu G$, the Larmor radius is 0.36 pc, which is small compared to the radius of a suitable SNR. However, if the magnetic field is amplified, this limit can be overcome. The paper investigates the possibility that the magnetic field in the acceleration region is amplified by the action of CR. Radio observations suggest that fields as large as $1 mG$ are present in some young SNR. The interaction of CR with the upstream plasma generates strong turbulence, which could wind up a large 'frozen-in' magnetic field. A non-linear theory of CR-driven turbulence is needed. The paper shows that for shocks with high Alfvén Mach number, the process by which streaming CR excite MHD turbulence is different from that usually supposed. This allows for a non-linear simulation with realistic parameters, showing that strong magnetic field amplification can be expected in young SNR. The paper also discusses the linear theory of magnetic field generation and the behavior of waves in different regimes. It shows that the maximum growth rate of waves occurs in a specific regime, and that the growth rate increases with the square root of the CR current in this regime. The paper concludes that the maximum attainable CR energy is limited by the condition that the acceleration time must be less than the age of the SNR. This is determined by the ratio of the Larmor radius to the SNR radius. The paper also discusses the importance of magnetic field amplification in overcoming this limit.