The electrical properties of flexible nonvolatile organic bistable devices (OBDs) fabricated with graphene sandwiched between two insulating poly(methyl methacrylate) (PMMA) polymer layers were investigated. Current-voltage (I-V) measurements on the Al/PMMA/graphene/PMMA/indium-tin-oxide/poly(ethylene terephthalate) devices at 300 K showed a current bistability due to the existence of the graphene, indicative of charge storage in the graphene. The maximum ON/OFF ratio of the current bistability for the fabricated OBDs was as large as 1 x 10(7), and the endurance number of ON/OFF switchings was 1.5 x 10(5) cycles, and an ON/OFF ratio of 4.4 x 10(6) was maintained for retention times larger than 1 x 10(5) s. No interference effect was observed for the scaled-down OBDs containing a graphene layer. The memory characteristics of the OBDs maintained similar device efficiencies after bending and were stable during repetitive bendings of the OBDs. The mechanisms for these characteristics of the fabricated OBDs are described on the basis of the I-V results.