In the field of spintronics, new phenomena and functions due to spin-orbit coupling have been discovered in recent years. The area based on spin-orbit coupling is particularly called spin-orbitronics. One of the essential technologies in spin-orbitronics is current-induced spin-orbit torques that enable the efficient manipulation of spintronic devices. Spin-orbit torques in various materials, such as metals and oxides, have been investigated to achieve a larger spin-orbit torque efficiency. Especially, 5d transition metals such as Ir and Pt have attracted particular attention because they can exhibit interesting electronic properties due to the strong spin-orbit coupling. In this study, we focus on spin-orbit torques in rhenium, which has been the subject of limited previous research among 5d transition metals. We measured current-induced spin-orbit torques for rhenium/ferromagnetic-metal bilayers using the spin-torque ferromagnetic resonance measurement. Based on the measurement results, we discuss damping-like torque efficiencies for the bilayers. We also discuss the phenomenon of current-induced magnetization switching due to the damping-like torques. These results will lead to the development of spin-orbitronics with a focus on spin-orbit torques.