Preserver problems have attracted considerable interest in linear algebra and functional analysis over the past decades. In the beginning, the attention was paid to descriptions of linear maps on spaces of matrices or operators that preserve certain numerical quantities, operations, relations, etc among the elements of the underlying spaces. This has been changed recently and now research focuses mainly on descriptions of transformations which are not necessarily linear, they may even be defined on non-linear structures.

In this talk we deal with preservers on positive (semidefinite or definite) matrices. We presents structural results concerning transformations on their sets which preserve certain

\item[-] algebraic operations (sorts of products, matrix means); \item[-] relations (different kinds of orders, commutativity); \item[-] numerical quantities (distances, divergences).

Interrelations between those problems will be discussed and some applications (e.g., ones related to physics) will be given. If time permits, we shall make comments on infinite dimensional generalizations in the setting of operator algebras.