Beta-detected nuclear spin relaxation of 8Li+ has been used to study important problems in polymer physics. In the first case we probe the depth dependence of molecular dynamics in high- and low-molecular-weight deuterated polystyrene (PS-d8). The average nuclear spin-lattice relaxation rate, 1/T1 avg, is a measure of the spectral density of the polymer dynamics at the Larmor frequency (41MHz at 6.55Tesla). The mean fluctuation rate decreases approximately exponentially with distance from the free surface, returning to bulk behavior for depths greater than ~10nm and the effective thickness of the surface region increases with increasing temperature. These results present challenges for the current understanding of dynamics near the free surface of polymer glasses. In the second case, we use the technique to make the first quantitative measurements of surface segregation in samples that are blends of two chemically identical polymers with different degrees of polymerization.