We derive formulae for the higher order tail moments of the lower truncated multivariate standard normal (MVSN), Student’s , lognormal and a finite-mixture of multivariate normal distributions (FMVN). For the MVSN we propose a recursive formula for moments of arbitrary order as a generalization of previous research. For the Student’s -distribution, the recursive formula is an extension of the normal case and when the degrees of freedom V→∞ the tail moments converge to the normal case. For the lognormal, we propose a general result for distributions in the positive domain. Potential applications include robust statistics, reliability theory, survival analysis and extreme value theory. As an application of our results we calculate the exceedance skewness and kurtosis and we propose a new definition of multivariate skewness and kurtosis using tensors with the moments in their components. The tensor skewness and kurtosis captures more information about the shape of distributions than previous definitions.