The Lu 1-xYb xNi 2B 2C system exhibits superconductivity at low Yb concentrations (0 x 0.1), Kondo behaviour at low and moderate Yb concentrations (0 x 0.34), and heavy fermion behaviour with a non-magnetic ground state at high Yb concentrations. In order to characterize the evolution of these phenomena with x , we have performed dc magnetic susceptibility, electrical resistivity, thermoelectric power, and specific heat measurements on single crystals of Lu 1- x Yb x Ni 2 B 2 with various values of x between 0 and 1. The enhanced suppression of T c by Yb substitution, compared to other rare earth substitutions, is consistent with the behaviour of superconducting Kondo systems with T K T c when T K decreases with increasing x . The electronic specific heat coefficient increases from 11 mJ/mol K 2 for x =0 to 530 mJ/mol K 2 for x =1. The curve of the reduced specific heat jump C / C 0 versus T c / T c0 , where C 0 and T c0 refer to the LuNi 2 B 2 C matrix, does not conform to the BCS theory, but is consistent with the magnetic pair breaking theory of Abrikosov and Gor'kov or the theory of Mller-Hartmann and Zittartz for superconductivity in the presence of the Kondo effect for . This is in agreement with the evolution of the upper critical field H c2 with x , where the slope decreases with increasing x.