NOVELTY - Preparing a polymer-modified substrate, comprises: (a) preparing on at least a part of the surface of the substrate a binding layer covalently attached to the substrate and carrying many vinyl groups substituted by an electron accepting group; (b) contacting the binding layer with rare earth metal complexes and allowing the rare earth metal complexes to coordinate with the vinyl groups of the binding layer; (c) contacting the binding layer including the coordinated rare earth metal complex catalysts with vinyl monomers; and (d) carrying out a polymerization of the vinyl monomers.    USE - The process is useful preparing a polymer-modified substrate, which is useful: to impart flame retardant properties, anti-fouling properties, or biocompatibility; as a substrate for the adhesion of cells, tissues and/or proteins; and for the expansion of cells, which are stem cells including embryonic and adult stem cells, pluripotent cells and mesenchymal stromal cells (all claimed); and as fuel cell membrane or on a fuel cell membrane.    ADVANTAGE - The process: provides a polymer-modified substrate that has excellent biocompatibility and/or anti-fouling properties; and increases the mechanical stability and the durability of the modified surface.    DETAILED DESCRIPTION - Preparing a polymer-modified substrate, comprises: (a) preparing on at least a part of the surface of the substrate a binding layer covalently attached to the substrate and carrying many vinyl groups substituted by an electron accepting group; (b) contacting the binding layer with rare earth metal complexes as catalysts and allowing the rare earth metal complexes to coordinate with the vinyl groups substituted by an electron accepting group of the binding layer; (c) contacting the binding layer including the coordinated rare earth metal complex catalysts with vinyl monomers containing a vinyl group substituted with an electron accepting group; and (d) carrying out a polymerization of the vinyl monomers mediated by the rare earth metal of the coordinated rare earth metal complex to form polymer chains covalently attached at one of their terminals to the binding layer. INDEPENDENT CLAIMS are also included for:    (1) the polymer modified substrate comprising a substrate, a binding layer covalently attached to the surface of the substrate and covering at least a part of this surface, and a polymer brush formed by many polymer chains each of which is covalently attached at one of its terminals to the binding layer; and    (2) a binding-layer modified substrate comprising a substrate and a binding layer covalently attached to the surface of the substrate and covering at least a part of this surface.