The Lu 1Àx Yb x Ni 2 B 2 C system exhibits superconductivity at low Yb concentrations ð0 x 0:1Þ, Kondo behaviour at low and moderate Yb concentrations ð0 x 0:34Þ, 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 1Àx Yb x Ni 2 B 2 C 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 ¼ 0 to $ 530 mJ/mol K 2 for x ¼ 1. The curve of the reduced specific heat jump ÁC=ÁC 0 versus T c =T c0 , where ÁC 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 Mu¨ller-Hartmann and Zittartz for superconductivity in the presence of the Kondo effect for T K =T c0 $ 10 3 . This is in agreement with the evolution of the upper critical field H c2 with x, where the slope dH c2 =dTj T¼Tc decreases with increasing x.