  Energetic nonthermal particles (cosmic rays, CRs) are accelerated in
supernova remnants, relativistic jets and other astrophysical objects. The CR
energy density is typically comparable with that of the thermal components and
magnetic fields. In this review we discuss mechanisms of magnetic field
amplification due to instabilities induced by CRs. We derive CR kinetic and
magnetohydrodynamic equations that govern cosmic plasma systems comprising the
thermal background plasma, comic rays and fluctuating magnetic fields to study
CR-driven instabilities. Both resonant and non-resonant instabilities are
reviewed, including the Bell short-wavelength instability, and the firehose
instability. Special attention is paid to the longwavelength instabilities
driven by the CR current and pressure gradient. The helicity production by the
CR current-driven instabilities is discussed in connection with the dynamo
mechanisms of cosmic magnetic field amplification.
