 Patients with non-valvular atrial fibrillation who are at risk for stroke and non-central nervous system non-CNS systemic embolism, will be randomized assigned by chance to receive treatment with rivaroxaban or warfarin, two different anticoagulants substances that prevent blood clots. Treatment will be double-blinded neither the patient nor study staff will know which study drug is assigned to patients during the study. Patients assigned to rivaroxaban will receive rivaroxaban 20 mg orally p.o. once daily OD plus warfarin placebo p.o. OD titrated to a target sham international normalized ratio INR of 2.5. Patients with moderate renal impairment at screening will receive rivaroxaban 15 mg p.o. OD. Patients assigned to warfarin will receive warfarin p.o. OD titrated to a target INR of 2.5 plus rivaroxaban placebo p.o. OD. The maximum expected length of treatment is up to 32 months but may be extended up to 4 years. 