In this paper we introduce a quite natural definition of exactness for penalty functions and we show that the best known class of nondifferentiable penalty functions is exact according to this definition. Moreover, we introduce a new class of nondifferentiable exact penalty functions containing a barrier term which causes the unconstrained minimizers to be located in the interior of a compact set; this allows the construction of an unconstrained algorithm which can be shown to be globally convergent towards K-T points of the constrained problem.