  We study gauge theories based on abelian $p-$forms on real compact hyperbolic
manifolds. An explicit formula for the conformal anomaly corresponding to
skew--symmetric tensor fields is obtained, by using zeta--function
regularization and the trace tensor kernel formula. Explicit exact and
numerical values of the anomaly for $p-$forms of order up to $p=4$ in spaces of
dimension up to $n=10$ are then calculated.
