This paper deals with multivariate Gaussian models for which the covariance matrix is a Kronecker product of two matrices. We consider maximum likelihood estimation of the model parameters, in particular of the covariance matrix. There is no explicit expression for the maximum likelihood estimator of a Kronecker product covariance matrix. We investigate whether the maximum likelihood estimator of the covariance matrix exists and whether it is unique. We consider models with general, with double diagonal, and with one diagonal Kronecker product covariance matrices, and find different results.