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This paper is concerned with the blind identification of Volterra-Hammerstein systems. Two identification scenarios are covered. The first scenario assumes that, although the input is not available, the statistics of the input are a priori known. This case appears in communication applications where the input statistics of the transmitter are known to the receiver. The second scenario assumes that the input statistics are unknown. In the case of known input statistics, the input is stationary higher order white noise with arbitrary probability density function. Under the scenario of unknown input statistics, the input is restricted to Gaussian white process. New cumulant-based identification methods are described for the above scenarios. The problem is converted into a linear multivariable form and the output cumulants are calculated using Kronecker products. First, initial conditions are determined by a linear system of equations. These correspond to the boundary values of the Volterra kernels. The remaining kernel coefficients can be determined under both identification schemes from a possibly overdetermined system of linear equations.