The direct dissocn. of CO on the Ru(0001) surface at two coverages (11.1% and 25.0%) is studied, as well as the dissocn. of CO on a stepped Ru(0001) surface. Five reaction paths are studied on the planar most compact surface and four on the stepped surfaces. The indirect route for CO dissocn. via the insertion of an H atom is presented for the planar surface, as well as the Boudouard reaction for the stepped surface.