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In multiple testing, strong control of the familywise error rate (FWER) may be unnecessarily stringent in some situations such as bioinformatic studies. An alterna-tive is to control the false discovery rate (FDR), the expected proportion of true null hypotheses among all rejected null hypotheses. However, in bioinformatic studies, the loss or cost of false discoveries often corresponds to the number rather than the propor-tion of false discoveries. Controlling the generalized familywise error rate (gFWER) controls the probability of incorrectly rejecting strictly more than m hypotheses. In this dissertation, we propose the generalized partitioning principle for constructing multiple tests that control gFWER. A set of sucient conditions to shortcut general-ized partitioning tests as step-down tests is provided. We show that, by being able to use information on the joint distribution of test statistics, step-down tests we propose can be more powerful than stepdown tests that ignore such information. ii This is dedicated to the memory of my grandmother, Xiula