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In this paper, we investigate the power of various normality tests against alternative distributions using Monte Carlo simulation experiments. We use seven different normality tests classified as moments tests, correlation and regression tests, and empirical distribution functional tests against six symmetric and four asymmetric alternative distributions. We also perform the rank analysis for the power of the normality tests. Furthermore, we conduct an empirical analysis of five emerging Asian stock indices (India, Indonesia, Malaysia, Singapore, and Taiwan) to understand whether the returns follow a normal distribution or not during the period from January 2000 to January 2020. We find that emerging Asian stock index returns do not follow normal distribution irrespective of the different frequencies of the data.