The authors review some classical and recent contributions concerning extensions, generalizations and variants of the Banach-Stone theorem on determination of the topology of a compact space
by the linear metric structure of
. The paper consists of five sections: introduction (a short description of the content), classical results, isometries, algebra isomorphisms, vector lattice isomorphisms. The paper is based on the lectures delivered by the second author in a Summer Course of the University of Cantabria, held in Laredo in August 2001.