Preserver problems concern the question of determining or describing the general form of all transformations of a given structure X which preserve ''something relevant'' for X that can be a

-- a quantity attached to the element of X, or

-- a distinguished set of elements of X, or

-- a given relation among the elements of X, or

-- a given operation on X, etc.

Such problems arrise in most parts of mathematics and they are systematically studied in matrix theory and, recently, also in operator theory under the name ''Preserver Problems''.

In this talk we will first give a general overview of the topic and then present several results concerning preservers on quantum structures (spaces of density operators, observables, and Hilbert space effects).