We have studied the electronic structure and the current-voltage (I-V) characteristics of one-dimensional InSe nanoribbons using the density functional theory combined with the nonequilibrium Green’s function method. Nanoribbons having bare or H-passivated edges of types zigzag (Z), Klein (K), and armchair (A) are taken into account. Edge states are found to play an important role in determining their electronic properties. Edges Z and K are usually metallic in wide nanoribbons as well as their hydrogenated counterparts. Transition from semiconductor to metal is observed in hydrogenated nanoribbons HZZH as their width increases, due to the strong width dependence of energy difference between left and right edge states. Nevertheless, electronic structures of other nanoribbons vary with the width in a very limited scale. The I-V characteristics of bare nanoribbons ZZ and KK show strong negative differential resistance, due to spatial mismatch of wave functions in energy bands around the Fermi energy. Spin polarization in these nanoribbons is also predicted. In contrast, bare nanoribbons AA and their hydrogenated counterparts HAAH are semiconductors. The band gaps of nanoribbons AA (HAAH) are narrower (wider) than that of two-dimensional InSe monolayer and increase (decrease) with the nanoribbon width.