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In this talk, we discuss anisotropic versions of Hardy inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for $L^{p}$-weighted Hardy inequalities on homogeneous (Lie) groups, which are also new in the Euclidean setting. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of $L^{p}$-weighted Hardy inequalities involving a distance and stability estimates. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. We also establish sharp Hardy type inequalities in $L^{p}$ with superweights. There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters. This talk is based on our recent book with Michael Ruzhansky "Hardy inequalities on homogeneous groups: 100 years of Hardy inequalities", Progress in Mathematics, Birkhauser, to appear.