Statisticians are interested in testing the structure of covariance matrices, especially under the high-dimensional scenario in which the dimensionality of data matrices exceeds the sample size. Many test statistics have been introduced to test whether the covariance matrix is equal to identity structure (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?H_%7B01%7D:%20%5CSigma%20=%20I_p" />), sphericity structure (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?H_%7B02%7D:%20%5CSigma%20=%20%5Csigma%5E2I_p" />) or diagonal structure (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?H_%7B03%7D:%20%5CSigma%20=%20diag(d_1,%20d_2,%5Cdots,d_p)" />). These test statistics work under the assumption that data follows the multivariate normal distribution. In our thesis work, we want to compare the performance of test statistics for each structure test under given assumptions and when the distributional assumption is violated, and compare the test sensitivity to outliers. We apply simulation studies with the help of significance level, power of test, and goodness of fit tests to evaluate the performance of structure test statistics. In conclusion, we identify the recommended test statistics that perform well under different scenarios. Moreover, we find out that the test statistics for the identity structure test are more sensitive to the changes of distribution assumptions and outliers compared with others. The test statistics for the diagonal structure test have a better tolerant to the change of the data matrices.