The structural and electronic properties of substitutional phosphorus and nitrogen impurities in ZnSe are studied using pseudopotential total-energy calculations. Substitutional phosphorus and nitrogen, in their neutral states, form shallow acceptors. In the case of phosphorus, the symmetry is lowered from ${\mathit{T}}_{\mathit{d}}$ to ${\mathit{C}}_{3\mathit{v}}$ by a Jahn-Teller distortion, and the hole is more localized on the phosphorus site. For nitrogen, however, the Jahn-Teller distortion is so small that the symmetry is approximately ${\mathit{T}}_{\mathit{d}}$, and the hole density is spread around nearby selenium-atom sites. For nitrogen in the positive charge state, two minimal-energy configurations are predicted: a stable state with approximately ${\mathit{T}}_{\mathit{d}}$ symmetry and a metastable state with ${\mathit{C}}_{3\mathit{v}}$ symmetry, which is 0.3 eV higher in energy than the stable one.