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The paper deals with theoretical and practical aspects of location parameter robust estimation for data samples of limited size under assumption that they obey a heavy-tailed distribution symmetric with respect to location parameter. It is assumed that distribution and/or its parameter are a priori unknown. Then, adaptation is desirable in order to provide appropriate accuracy. Four different adaptive robust estimators are considered. Their accuracy is studied and compared between each other as well as to accuracy that can be provided by maximum-likelihood and optimal L-estimators. It is shown that the considered adaptive estimators perform well for three typical families of non-Gaussian distributions widely used to model non-Gaussian heavy-tailed processes (data).