This paper is concerned with the state estimation problem for two-dimensional (2D) complex networks with randomly occurring nonlinearities and randomly varying sensor delays. To describe the fact that measurement delays may occur in a probabilistic way, the randomly varying sensor delays are introduced in the delayed sensor measurements. The randomly occurring nonlinearity, on the other hand, is included to account for the phenomenon of nonlinear disturbances appearing in a random fashion that is governed by a Bernoulli distributed white sequence with known conditional probability. The stochastic Brownian motions are also considered, which enter into not only the coupling terms of the complex networks but also the measurements of the output systems. Through available actual network measurements, a state estimator is designed to estimate the true states of the considered 2D complex networks. By utilizing an energy-like function, the Kronecker product and some stochastic analysis techniques, several sufficient criteria are established in terms of matrix inequalities under which the 2D estimation error dynamics is globally asymptotically stable in the mean square. Furthermore, the explicit expression of the estimator gains is also characterized. Finally, a numerical example is provided to demonstrate the effectiveness of the design method proposed in this paper. Copyright © 2012 John Wiley & Sons, Ltd.