The single machine scheduling problem to minimize the number of tardy jobs is one of the most basic and important scheduling problems in classical scheduling theory. An algorithm of Moore, which is sometimes known as the Moore-Hodgson algorithm, solves the problem in O (n log n) time. This algorithm repeatedly adds jobs in EDD order to the end of a partial schedule of on-time jobs. If the addition of job j results in this job completed after time dj, then a job in the partial schedule with the largest processing time is removed and declared late. Discarding a job with the largest processing time increases the opportunity for subsequent jobs to be completed by their due dates. In the past 40 years, lots of researchers studied it with ever-increasing interests and profound results were appeared continually. For example, Sidney generalizes this algorithm to handle the case that a specified subset of jobs must be on time. He observes that jobs of the specified subset are not considered when discarding jobs, and that it may be necessary to discard more than one job to ensure that the last job in the current partial schedule is on time. Adaptations of Moore-Hodgson’s algorithm to both problem 1|(ri≤rj)?(di≤dj)|∑Uj and problem 1|(pi≤pj)?(wi≥wj)|∑wjUj for the cases of both agreeability of release dates with due dates, and reverse agreeability of processing times with weights are proposed by both Kise, Ibaraki and Mine, and Lawler, respectively. In this paper we review results and their significances on the problems obtained by Master students in Chongqing Normal University, including classical ones and their generalizations.