In this paper, it is shown that a complex multivariate random variable z = (Z1, Z2,…,Zp)' is a complex multivariate normal random variable of dimensionality p if and only if all nondegenerate complex linear combinations of Z have a complex univariate normal distribution. The characteristic function of Z has been derived, and simpler forms of some theorems have been given using this characterization theorem without assuming that the variance-covariance matrix of the vector Z is Hermitian positive definite. Marginal distributions of Z have been given. In addition, a complex multivariate t-distribution has been defined and the density derived. A characterization of the complex multivariate t-distribution is given. A few possible uses of this distribution have been suggested.