  Particle acceleration in plasma creates a possibility of exceptionally high
accelerating gradients and appears as a very attractive option for future
linear electron-positron and/or photon-photon colliders. These high
accelerating gradients were already demonstrated in a number of experiments.
However, a linear collider requires exceptionally high beam brightness which
still needs to be demonstrated. In this article we discuss major phenomena
which limit the beam brightness of accelerated beam and, consequently, the
collider luminosity.
