Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces Over the past several decades the territory of preserver problems has been continuously enlarging within the frame of linear analysis. The aim of this work is to present a sort of cross section of the modern theory of preservers on infinite dimensional spaces (operator spaces and function spaces) through the authors corresponding results. Special emphasis is put on preserver problems concerning some structures of Hilbert space operators which appear in quantum mechanics. Moreover local automorphisms and local isometries of operator algebras and function algebras are discussed in details. Some Linear and Multiplicative Preserver Problems on Operator Algebras and Function Algebras Preservers on Quantum Structures Local Automorphisms and Local Isometries of Operator Algebras and Function Algebras.