The problem of testing a null hypothesis H about a multivariate distribution for simple as well as for composite H is considered. H will be rejected when u, the observed value of a real-valued test statistic U is above a specified constant, which cannot be evaluated from the distribution function F of U. If it is possible, by using pseudo-random numbers, to simulate a random sample $u\sb 2,...,u\sb n$ of n-1 observations from F, a test can be constructed. If F is absolutely continuous and if u is the k th largest of the values $u,u\sb 2,...,u\sb n$, then H will be rejected at the k/n significance level. Such a test is called a Monte Carlo test (MCT).
The paper describes two methods of constructing MCTs for cases where the independence assumption for random samples is violated. The methods are applied to the Rasch model, to the finite lattice Ising model and to testing of association between spatial processes.