The present invention relates to formaldehyde-free aqueous binders for substrates, having a broad molecular weight distribution and comprising (A) from 0 to 100% by weight of an ethylenically unsaturated acid anhydride or ethylenically unsaturated dicarboxylic acid whose carboxylic acid groups can form an anhydride group, or mixtures thereof, (B) from 100 to 0% by weight of an ethylenically unsaturated compound, (C) at least one polyfunctional crosslinker or mixtures thereof, the polymers of A) and B) obtained by free-radical addition polymerization, when classified in a coordinate system by way of their average molecular weight Mw and their polydispersity, being situated in the area above a straight line which is defined by the linear equation y=1.25x+20 000 and has been shifted in y direction parallelwise by at least +3 000, the x axis denoting the weight-average molecular weight and the y axis the polydispersity times 10 000.