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The problem of adaptive robust state observer design is considered for a class of uncertain nonlinear systems with delayed state perturbations. It is supposed that the upper bound of the nonlinearity and uncertainty, including delayed states, is a linear function of some parameters which are still assumed to be unknown. Here, it is not required that the nonlinear terms including delayed states are linear norm-bounded in the states. An improved adaptation law with sigma-modification is employed to estimate the unknown parameters. Then, by making use of the updated values of these unknown parameters a class of memoryless adaptive robust state observers is proposed for uncertain nonlinear time-delay systems. It is also shown that by employing the proposed adaptive robust state observer, the observation error between the observer state estimate and the true state can converge asymptotically to zero in the presence of significant uncertainties and time delays. Finally, a numerical example is given to demonstrate the validity of the results.