To date, nonlinear Linfinity-gain filtering problems have not been solved by conventional methods for nonlinear dynamic systems with persistent bounded disturbances. This study introduces a fuzzy filtering design to deal with the nonlinear Linfinity-gain filtering problem. First, the Takagi and Sugeno fuzzy model is employed to approximate the nonlinear dynamic system. Next, based on the fuzzy model, a fuzzy filter is developed to minimize the upper bound of Linfinity-gain of the estimation error under some linear matrix inequality (LMI) constraints. Therefore, the nonlinear Linfinity-gain filtering problem is transformed into a suboptimal filtering problem, i.e., to minimize the upper bound of the Linfinity-gain of the estimation error subject to some LMI constraints. In this situation, the nonlinear Linfinity-gain filtering problem can be easily solved by an LMI-based optimization method. The proposed methods, which efficiently attenuate the peak of estimation error due to persistent bounded disturbances, extend the Linfinity-gain filtering problems from linear dynamic systems to nonlinear dynamic systems.