Jordan isomorphisms as preservers

This article is an extended version of the talk delivered by the author at The International Conference on Algebra and Related Topics (ICART) 2018 in Rabat, Morocco. In the first part we discuss certain preserver properties of Jordan *-isomorphisms on C*-algebras and present some classical results showing that, to some extents, those properties in fact characterize Jordan *-isomorphisms among bijective linear transformations. In the second part we survey some of our recent results which can be regarded as non-linear characterizations of Jordan *-isomorphisms on the cones of positive elements in operator algebras and on the spaces of self-adjoint elements. Besides, we give a few new results which extend or unify certain former statements that have appeared in the literature. Finally, we also pose some open problems for further research.