In this thesis, the topic is concentrated on FDR control proposed by Benjamini and Hochberg (1995) and a Cram´er moderate deviation theorem established by Liu, W. and Shao, Q. M.(2013). Benjamini and Hochberg proposed the false discovery rate (FDR) in multiple testing problems. Compared with FWER control, FDR control is a more efficient and powerful. After that, researchers showed great passions on methods of FDR control in various models. In this thesis, we also review other three popular methods and give a comparison of these methods by analyzing a real experimental data. As a natural development of multiple testing problems, we talk about a Cram´er moderate deviation theorem for Hotelling’s T 2−statistic established by Liu, W. and Shao, Q. M.(2013). The result builds a bridge for us to exactly know the number of tests in which χ 2−distribution could be used to estimate p−values within the scope of test accuracy.