We present a detailed study of RNi1−xBi2±y (R = La–Nd, Sm, Gd–Dy) single crystals by measurements of stoichiometry and temperature dependent magnetic susceptibility, magnetization, and electrical resistivity. This series forms with partial Ni occupancy, 0.72 ⩽ (1−x) ⩽ 0.84, as well as a variable Bi occupancy, 1.76 ⩽ (2±y) ⩽ 2.14. For R = Ce–Nd, Gd–Dy, the RNi1−xBi2±y compounds show local-moment like behavior and order antiferromagnetically at low temperatures. Determination of anisotropies as well as antiferromagnetic ordering temperatures for RNi1−xBi2±y (R = Ce–Nd, Sm, Gd–Dy) have been made. Crystalline samples from this family exhibit minority, second phase superconductivity at low temperatures, which can be associated with Ni–Bi and Bi contamination. No evidence of bulk superconductivity has been observed.