This paper investigates the boundedness and stability of solutions to the affine variational inequality problem. The concept of a solution ray to a variational inequality defined by an affine mapping and on a closed convex set is introduced and characterized; the connection of such a ray with the boundedness of the solution set of the given problem is explained. In the case of the monotone affine variational inequality, a complete description of the solution set is obtained which leads to a simplified characterization of the boundedness of this set as well as to a new error bound result for approximate solutions to such a variational problem. The boundedness results are then combined with certain degree-theoretic arguments to establish the stability of the solution set of an affine variational inequality problem.