We report unipolar resistive switching in ultrathin films of chemically produced graphene (reduced graphene oxide) and multiwalled carbon nanotubes. The two-terminal devices with yield >99% are made at room temperature by forming continuous films of graphene of thickness ∼20 nm on indium tin oxide coated glass electrode, followed by metal (Au or Al) deposition on the film. These memory devices are nonvolatile, rewritable with ON/OFF ratios up to ∼ 10 5 and switching times up to 10 μs. The devices made of MWNT films are rewritable with ON/OFF ratios up to ∼400. The resistive switching mechanism is proposed to be nanogap formation and filamentary conduction paths.