  Emittance is a beam quality that is vital for many future applications of
advanced accelerators, such as compact free-electron lasers and linear
colliders. In this paper, we review the challenges of preserving the transverse
emittance during acceleration, both inside and outside accelerator stages.
Sources of emittance growth range from space charge and instabilities caused by
transverse wakefields, which can occur in any advanced accelerator scheme
regardless of medium or driver type, to sources more specific to plasma
accelerators, such as mismatching, misalignment, ion motion, Coulomb
scattering, chromaticity between stages, and more.
