  The article is a survey related to a classical unsolved problem in Banach
space theory, appearing in Banach's famous book in 1932, and known as the Mazur
rotations problem. Although the problem seems very difficult and rather
abstract, its study sheds new light on the importance of norm symmetries of a
Banach space, demonstrating sometimes unexpected connections with renorming
theory and differentiability in functional analysis, with topological group
theory and the theory of representations, with the area of amenability, with
Fra\"iss\'e theory and Ramsey theory, and led to development of concepts of
interest independent of Mazur problem. This survey focuses on results that have
been published after 2000, stressing two lines of research which were developed
in the last ten years. The first one is the study of approximate versions of
Mazur rotations problem in its various aspects, most specifically in the case
of the Lebesgue spaces Lp. The second one concerns recent developments of
multidimensional formulations of Mazur rotations problem and associated
results. Some new results are also included.
