This paper surveys some of the breakthroughs in the fast emerging theory of, what we will term, structure morphisms. Unlike the, by now classical, linear preservers, the structure morphisms focus on a given structure alone, without linearity imposed. For example, one can consider the set of invertible matrices and build a structure of a group. Its structure morphisms are usually called group homomorphisms.
The author of the paper under review discusses six, seemingly unrelated, problems of this type. The motivation for studying them usually comes from physics (relativity/quantum mechanics) or geometry, and is given in the introduction that precedes a given problem. The structure morphisms are then revealed, and well selected open problem(s) follow as an endnote.
The author concludes the survey by showing the intimate connection between the six discussed problems.