We introduce several mechanisms to dissipate the energy in the Benjamin-Bona-Mahony (BBM) equation. We consider either a distributed (localized) feedback
law, or a boundary feedback law. In each case, we prove the global wellposedness
of the system and the convergence towards a solution of the BBM equation which is
null on a band. If the Unique Continuation Property holds for the BBM equation, this
implies that the origin is asymptotically stable for the damped BBM equation.