The numerical availability of distribution functions is of fundamental importance for statistical computations. Due to the evaluation of multidimensional integrals, the numerical computation of multivariate distribution functions is much more complex than that of univariate distributions. This is particularly true for the indispensable multivariate t and normal distributions. They occur frequently in such different
application areas as Bayesian computations, multiple comparisons procedures, econometrics, integrated log-likelihood problems, and asymptotic theory in general.

Methods for the numerical computation of multivariate normal and multivariate t probabilities will be reviewed. The focus of the talk will be a description of several effective methods for problems that arise in important practical applications, along with some analysis of the computational complexity of these methods.