The unscented transform uses a weighted set of samples called sigma points to propagate the means and covariances of nonlinear transformations of random variables However, unscented transforms developed using either the Gaussian assumption or a minimum set of sigma points typically fall short when the random variable is not Gaussian distributed and the nonlinearities are substantial In this paper, we develop the generalized unscented transform (GenUT), which uses adaptable sigma points that can be positively constrained, and accurately approximates the mean, covariance, and skewness of an independent random vector of most probability distributions, while being able to partially approximate the kurtosis For correlated random vectors, the GenUT can accurately approximate the mean and covariance In addition to its superior accuracy in propagating means and covariances, the GenUT uses the same order of calculations as most unscented transforms that guarantee third-order accuracy, which makes it applicable to a wide variety of applications, including the assimilation of observations in the modeling of the coronavirus (SARS-CoV-2) causing COVID-19