In a collinear beam-driven wakefield accelerator, a bunch of
charged particles is accelerated by a strong electric field that is
generated in a medium by a preceding high-charge drive
bunch. Multiple beam-driven acceleration concepts have been proposed
and demonstrated in proof-of-principle experiments. In some
concepts, the medium is plasma where very strong electric fields are
created due to the motion of ions and electrons with respect to each
other. In other configurations, the medium is a slow-wave
electromagnetic structure made of dielectric and/or metal, and high
gradients are achieved due to the very short duration of the
electromagnetic pulse excited in the structure by the drive
bunch. Because of the high charge, and consequently long length of
the drive bunch, wakefields excited by the leading particles of the
drive bunch affect the trailing particles in the same bunch and
result in beam-driven instabilities obstructing the drive bunch's
stable propagation and extended interactions with the witness bunch,
ultimately terminating the energy transfer process. This paper
presents an overview of the drive-bunch beam dynamics in beam-driven
structure- and plasma-based accelerators with a focus on beam
instabilities that limit stable propagation of the drive bunch, such
as the beam break-up instability and transverse defocusing and
deflection in cases of cylindrical and planar structures and plasma
waveguides. Possible mitigation techniques are discussed.