We study the convergence of a class of penalty and barrier methods for solving monotone variational inequalities with constraints. This class of methods is an extension of penalty and barrier methods for convex optimization to the setting of variational inequalities. Primal convergence is established under weaker conditions than usual: the solution set is supposed either to be a compact set or, for the case of interior barrier methods, the sum of a compact set and a linear space. Dual convergence is also analyzed. The analysis strongly exploits a new formula related to recession calculus.