Two-dimensional crystals with a wealth of exotic dimensional-dependent properties are promising candidates for next-generation ultrathin and flexible optoelectronic devices. For the first time, we demonstrate that few-layered InSe photodetectors, fabricated on both a rigid SiO2/Si substrate and a flexible polyethylene terephthalate (PET) film, are capable of conducting broadband photodetection from the visible to near infrared region (450-785 nm) with high photoresponsivities of up to 12.3 AW-1 at 450 nm (on SiO2/Si) and 3.9 AW-1 at 633 nm (on PET). These photoresponsivities are superior to those of other recently reported 2D crystals (graphene, MoS2, GaS, and GaSe)-based photodetectors. The InSe devices fabricated on rigid SiO2/Si substrates possess a response time of ~50 ms and exhibit long-term stability in photo-switching. These InSe devices can also operate on a flexible substrate with or without bending, and reveal comparable performance to those devices on SiO2/Si. With these excellent optoelectronic merits, we envision that the nanoscale InSe layers will not only find applications in flexible optoelectronics, but also act as an active component to configure versatile 2D heterostructure devices.