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The article is a survey related to a classical unsolved problem in Banachspace theory, appearing in Banach's famous book in 1932, and known as the Mazurrotations problem. Although the problem seems very difficult and ratherabstract, its study sheds new light on the importance of norm symmetries of aBanach space, demonstrating sometimes unexpected connections with renormingtheory and differentiability in functional analysis, with topological grouptheory and the theory of representations, with the area of amenability, withFra\"iss\'e theory and Ramsey theory, and led to development of concepts ofinterest independent of Mazur problem. This survey focuses on results that havebeen published after 2000, stressing two lines of research which were developedin the last ten years. The first one is the study of approximate versions ofMazur rotations problem in its various aspects, most specifically in the caseof the Lebesgue spaces Lp. The second one concerns recent developments ofmultidimensional formulations of Mazur rotations problem and associatedresults. Some new results are also included.

On Mazur rotations problem and its multidimensional versions