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This paper addresses the issue of the convergence dynamics of stochastic Cohen-Grossberg neural networks (SCGNNs) with white noise, whose state variables are described by stochastic nonlinear integro-differential equations. With the help of Lyapunov functional, semi-martingale theory, and inequality techniques, some novel sufficient conditions on pth moment exponential stability and almost sure exponential stability for SCGNN are given. Furthermore, as byproducts of our main results, some sufficient conditions for checking stability of deterministic CGNNs with unbounded distributed delays have been established. Especially, even when the spectral radius of the coefficient matrix is greater than 1, in some cases our theory is also effective.