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We investigate state estimation for a class of discrete-time recurrent neural networks with leakage delay and time-varying delay. The design method for the state estimator to estimate the neuron states through available output measurements is given. A novel delay-dependent sufficient condition is obtained for the existence of state estimator such that the estimation error system is globally asymptotically stable. Based a novel double summation inequality and reciprocally convex approach, an improved stability criterion is obtained for the error-state system. Two numerical examples are given to demonstrate the effectiveness of the proposed design methods. The simulation results show that the leakage delay has a destabilizing influence on a neural network system. [ABSTRACT FROM AUTHOR]