This paper examines two fuzzy flowshop scheduling problems with fuzzy due dates. The membership function of a fuzzy due date assigned to each job represents the grade of satisfaction of a decision maker for the completion time of that job. One fuzzy flowshop scheduling problem is to maximize the minimum grade of satisfaction over given jobs, and the other is to maximize the total grade of satisfaction. First, we investigate the relations between the fuzzy scheduling problems and conventional scheduling problems. We next apply a genetic algorithm and neighborhood search algorithms (multi-start descent, taboo search and simulated annealing) to the fuzzy flowshop scheduling problems in order to examine the ability of these algorithms to find near optimal solutions. By computer simulations, it is shown that the maximization problem of the total satisfaction grade is even more tractable by these algorithms than that of the minimum satisfaction grade. Then, we show how the performance of these algorithms can be improved for the latter problem by utilizing problem domain knowledge. Finally, high performance of a hybrid genetic algorithm with neighborhood search is demonstrated.