During the last decade, many novel approaches for addressing multiplicity problems arising in clinical trials have been introduced in the literature. These approaches provide great flexibility in addressing given clinical trial objectives and yet maintain strong control of the familywise error rate. In this tutorial article, we review multiple testing strategies that are related to the following: (a) recycling local significance levels to test hierarchically ordered hypotheses; (b) adapting the significance level for testing a hypothesis to the findings of testing previous hypotheses within a given test sequence, also in view of certain consistency requirements; (c) grouping hypotheses into hierarchical families of hypotheses along with recycling the significance level between those families; and (d) graphical methods that permit repeated recycling of the significance level. These four different methodologies are related to each other, and we point out some connections as we describe and illustrate them. By contrasting the main features of these approaches, our objective is to help practicing statisticians to select an appropriate method for their applications. In this regard, we discuss how to apply some of these strategies to clinical trial settings and provide algorithms to calculate critical values and adjusted p-values for their use in practice. The methods are illustrated with several numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.