  Diffusive shock acceleration is the theory of particle acceleration through
multiple shock crossings. In order for this process to proceed at a rate that
can be reconciled with observations of high-energy electrons in the vicinity of
the shock, and for cosmic rays protons to be accelerated to energies up to
observed galactic values, significant magnetic field amplification is required.
In this review we will discuss various theories on how magnetic field
amplification can proceed in the presence of a cosmic ray population. On both
short and long scales, cosmic ray streaming can induce instabilities that act
to amplify the magnetic field. Developments in this area that have occurred
over the past decade are the main focus of this paper.
