An LMI approach is proposed for the design of robust H-infinity observers for a class of Lipschitz nonlinear systems. Two type of systems are considered, Lipschitz nonlinear discrete-time systems and Lipschitz nonlinear sampled-data systems with Euler approximate discrete-time models. Observer convergence when the exact discrete-time model of the system is available is shown. Then, practical convergence of the proposed observer is proved using the Euler approximate discrete-time model. As an additional feature, maximizing the admissible Lipschitz constant, the solution of the proposed LMI optimization problem guaranties robustness against some nonlinear uncertainties. The robust H-infinity observer synthesis problem is solved for both cases. The maximum disturbance attenuation level is achieved through LMI optimization. (C) 2007 Published by Elsevier Ltd.