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The two-dimensional (2D) C3N has emerged as a material with promising applications in high performance device owing to its intrinsic bandgap and tunable electronic properties. Although there are several reports about the bandgap tuning of C3N via stacking or forming nanoribbon, bandgap modulation of bilayer C3N nanoribbons (C3NNRs) with various edge structures is still far from well understood. Here, based on extensive first-principles calculations, we demonstrated the effective bandgap engineering of C3N by cutting it into hydrogen passivated C3NNRs and stacking them into bilayer heterostructures. It was found that armchair (AC) C3NNRs with three types of edge structures are all semiconductors, while only zigzag (ZZ) C3NNRs with edges composed of both C and N atoms (ZZ-CN/CN) are semiconductors. The bandgaps of all semiconducting C3NNRs are larger than that of C3N nanosheet. More interestingly, AC-C3NNRs with CN/CN edges (AC-CN/CN) possess direct bandgap while ZZ-CN/CN have indirect bandgap. Compared with the monolayer C3NNR, the bandgaps of bilayer C3NNRs can be greatly modulated via different stacking orders and edge structures, varying from 0.43 eV for ZZ-CN/CN with AB'-stacking to 0.04 eV for AC-CN/CN with AA-stacking. Particularly, transition from direct to indirect bandgap was observed in the bilayer AC-CN/CN heterostructure with AA'-stacking, and the indirect-to-direct transition was found in the bilayer ZZ-CN/CN with AB-stacking. This work provides insights into the effective bandgap engineering of C3N and offers a new opportunity for its applications in nano-electronics and optoelectronic devices.