A synthetic grass structure comprises a sheet substrate with a plurality of filiform formations extending from the substrate so as to simulate natural grass cover and a particulate filling material or infill consisting of a substantially homogeneous mass of a granular plastic material dispersed between the filiform formations so as to keep them in a substantially upright condition. The infill forms a layer of thickness not greater than 40 millimeters, has a grain size smaller than 5 millimeters, and consists of plastic material with a value of elastic return such that the structure has a KA value of 35-50, which enables its use for making athletics tracks.