The conditions for uniqueness of solutions to the barotropic vorticity equation within a limited region are discussed, in particular for cases with flow through the boundary, when no physical boundary conditions exist. Two different sets of boundary conditions are given, for which the solution will remain uniquely defined as long as certain of its derivatives are bounded. A small perturbation on the initial solution will then also remain small, and the problem is thus properly posed. It is furthermore shown that similar conclusions may be drawn for the finite-difference vorticity equation of the “leapfrog” type, based on the symmetric-conservative Jacobian suggested by Arakawa and the normal five- or nine-point Laplacian, if in addition two stability conditions are satisfied, one of them being essentially the condition suggested by Charney, Fjörtoft, and von Neumann.