We have performed a comprehensive fi rst-principles study of the electronic and magnetic properties of two-dimensional (2D) transition-metal dichalcogenide (TMD) heterobilayers MX 2 /MoS 2 (M ¼ Mo, Cr, W, Fe, V; X ¼ S, Se). For M ¼ Mo, Cr, W; X ¼ S, Se, all heterobilayers show semiconducting characteristics with an indirect bandgap with the exception of the WSe 2 /MoS 2 heterobilayer which retains the direct-bandgap character of the constituent monolayer. For M ¼ Fe, V; X ¼ S, Se, the MX 2 /MoS 2 heterobilayers exhibit metallic characters. Particular attention of this study has been focused on engineering the bandgap of the TMD heterobilayer materials via application of either a tensile strain or an external electric fi eld. We fi nd that with increasing either the biaxial or uniaxial tensile strain, the MX 2 /MoS 2 (M ¼ Mo, Cr, W; X ¼ S, Se) heterobilayers can undergo a semiconductor-to-metal transition. For the WSe 2 / MoS 2 heterobilayer, a direct-to-indirect bandgap transition may occur beyond a critical biaxial or uniaxial strain. For M ( ¼ Fe, V) and X ( ¼ S, Se), the magnetic moments of both metal and chalcogen atoms are enhanced when the MX 2 /MoS 2 heterobilayers are under a biaxial tensile strain. Moreover, the bandgap of MX 2 /MoS 2 (M ¼ Mo, Cr, W; X ¼ S, Se) heterobilayers can be reduced by the vertical electric fi eld. For two heterobilayers MSe 2 /MoS 2 (M ¼ Mo, Cr), PBE calculations suggest that the indirect-to-direct bandgap transition may occur under an external electric fi eld. The transition is attributed to the enhanced spontaneous polarization. The tunable bandgaps in general and possible indirect – direct bandgap transitions due to tensile strain or external electric fi eld make the TMD heterobilayer materials a viable candidate for optoelectronic applications.