The structure and electronic and magnetic properties of SnSe monolayer doped with transition-metal (TM) atoms (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) were investigated using first-principles calculations. It is found that TM atoms can substitute either Se atoms (TM-Se) or Sn atoms (TM-Sn) to form thermodynamically stable systems. Semiconductor–metal or semiconductor–half metal transitions may be induced by TM atom doping, depending on the types of dopants and substituted atoms. The pristine nonmagnetic SnSe monolayers can be turned into magnetic systems by TM dopants. Especially, for TM-Se systems (TM = Sc, V, Mn and Fe) and TM-Sn systems (TM = V, Cr, Mn, Fe and Co), the monolayers become two-dimensional narrow-band diluted magnetic semiconductors (DMSs). Among the magnetic systems, TM-Se systems can exhibit either obviously enhanced (TM = Sc, Ti, V and Cu) or weakened (TM = Mn, Fe and Co) total magnetic moments, due to the strong interaction between the 3d-orbit of the dopant atoms and the 5p-orbit of the neighboring Sn atoms, while TM-Sn systems only exhibit slightly enhanced total magnetic moments, which shows the greater electronic tunability of Sn atoms than that of Se atoms. Our results may provide a platform for potential applications of SnSe monolayer-based spintronic materials.