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In the Present Book Chapter-I is an introductory one. It contains the general introduction about the problem of nonnormal disturbances in linear statistical models, Chapter-II deals with the consequences of nonnormal disturbances in linear statistical models under finite and infinite variances of disturbances and It explains a few Robust estimators.Chapter-III describes the review about the various existing tests for normality of observations. It deals with Shapiro-Wilk 'W' test for normality and it's extensions along with the comparative study of various statistical procedures for evaluating the normality of a complete sample. Chapter - IV proposes some new test procedures for testing the normality of disturbances in linear statistical models by using the various types of residuals namely, studentized, predicted, recursive and Best Linear Unbiased Scalar (BLUS) residuals.Chapter -V presents the conclusions.Several selected references have been documented under a separate caption'BIBLIOGRAPHY'.