In this paper, we present a nonlinear state estimation algorithm based on the fusion of an extended H-infinity (EH infinity F) and a cubature Kalman filter (CKF); the resulting estimator is called a cubature H-infinity filter. The recently developed CKF is a Gaussian approximation of a Bayesian filter and its performance over non-Gaussian noises may degrade. In contrast, the H-infinity filter is capable of estimating the states of linear systems with non-Gaussian noises and the extended H-infinity filter (EH infinity F) can estimate the states of non-linear and non-Gaussian systems. Similar to the H-infinity filter, an EH infinity F also does not make any assumptions about the statistics of the process or measurement noise, but it does require Jacobians during the state estimation of nonlinear systems, which degrade the overall performance when the nonlinearities are severe. The cubature H-infinity filter is developed to have the desirable features of both CKF and EH infinity F. For numerical accuracy, a square-root version of the cubature H-infinity filter is developed using J-unitary transformation. The efficacy of the square-root cubature H-infinity filter is verified on continuous stirred tank reactor and permanent magnet synchronous motor examples.