Formation and characterization of graphene p-n junctions are of particular interest because the p-n junctions are used in a wide variety of electronic/photonic systems as building blocks. Graphene p-n junctions have been previously formed by using several techniques, but most of the studies are based on lateral-type p-n junctions, showing no rectification behaviors. Here, we report a new type of graphene p-n junction. We first fabricate and characterize vertical-type graphene p-n junctions with two terminals. One of the most important characteristics of the vertical junctions is the asymmetric rectifying behavior showing an on/off ratio of ~10(3) under bias voltages below ±10 V without gating at higher n doping concentrations, which may be useful for practical device applications. In contrast, at lower n doping concentrations, the p-n junctions are ohmic, consistent with the Klein-tunneling effect. The observed rectification results possibly from the formation of strongly corrugated insulating or semiconducting interlayers between the metallic p- and n-graphene sheets at higher n doping concentrations, which is actually a structure like a metal-insulator-metal or metal-semiconductor-metal tunneling diode. The properties of the diodes are almost invariant even 6 months after fabrication.