In this paper we use a penalty method to approximate a number of stopping problems over a finite horizon. In particular we prove existence and continuity of the value function corresponding to Dynkin games over a finite horizon. Since stopping problems can be studied in the context of Dynkin games, as a by-product we obtain continuity of the optimal stopping value. We then study Dynkin games with delayed stopping and finally impulse control. In each case the value function is approximated by a solution to a suitable penalty equation.