High-temperature spin dynamics in one-dimensional, spin 1/2, Heisenberg systems CuCl 2 ·2NC 5 H 5 and Cu(C 6 H 5 COO) 2 ·3H 2 O is studied by the measurements of proton spin-lattice relaxation rate, T -1 1 . The field dependence of T -1 1 can be fitted by an expression T -1 1 = P H -1/2 + Q , which gives direct evidence of spin diffusion in spin 1/2 linear chains. Additional evidence is presented which shows that a classic one-dimensional salt Cu(NH 3 ) 4 SO 4 ·H 2 O, in which one-dimensionality is now sceptical, also has one-dimensional spin correlations.