Versatile applications have driven a desire for dual-band detection that enables seeing objects in multiple wavebands through a single photodetector. In this paper, a concept of using graphene/p-GaN Schottky heterojunction on top of a regular AlGaN-based p-i-n mesa photodiode is reported for achieving solar-/visible-blind dual-band (275 nm and 365 nm) ultraviolet photodetector with high performance. The highly transparent graphene in the front side and the polished sapphire substrate at the back side allows both top illumination and back illumination for the dual band detection. A system limit dark current of 1×10-9 A/cm2 at a negative bias voltage up to -10 V has been achieved, while the maximum detectivity obtained from the detection wavebands of interests at 275 nm and 365 nm are ∼ 9.0 ×1012 cm·Hz1/2/W at -7.5 V and ∼8.0 × 1011 cm·Hz1/2/W at +10 V, respectively. Interestingly, this new type of photodetector is dual-functional, capable of working as either photodiode or photoconductor, when switched by simply adjusting the regimes of bias voltage applied on the devices. By selecting proper bias, the device operation mode would switch between a high-speed photodiode and a high-gain photoconductor. The device exhibits a minimum rise time of ∼210 µs when working as a photodiode and a maximum responsivity of 300 A/W at 6 μW/cm2 when working as a photoconductor. This dual band and multi-functional design would greatly extend the utility of detectors based on nitrides.