Five substitutional donors have been observed in ZnSe: Al, Ga, In, Cl, and F. By measuring the I2 lines and the two-electron transitions associated with each, the donor binding energies have been determined. These are found to be close to the effective-mass value and vary from 26.3 meV for Al to 29.3 meV for F. Excited states of the complex formed by the exciton bound to the neutral donor were observed both in the region above the I2 lines and in the two-electron transitions. The electron effective mass was measured from the Zeeman splitting of the 2p states of the donors to be m=(0.16+/-0.01)me. For each donor, a doublet was also observed at lower energy than the I2 lines; these doublets are believed to be the corresponding I3 lines. The binding energies of excitons both to the ionized and the neutral donors were found to vary linearly with the donor central-cell correction. Most of these results are closely analogous to the properties of CdS and CdSe.