  We study the $L^2$--cohomology of certain local systems on non-compact
arithmetic ball quotients $X=\Gamma \backslash \B_n$, in particular vanishing
and non--vanishing results. We also give generalizations to higher dimensional
ball quotients and study the mixed Hodge structure on the sheaf cohomology of a
local system with the $L^2$-cohomology contributing to the lowest weight part.
