Testing uniformity in the univariate case has been studied by many researchers. Many papers have been published on this issue, whereas the multi-dimensional uniformity test seems to have received less attention in the literature. A new test statistic for the multi-dimensional uniformity is proposed in this thesis. The proposed test statistic can be used to test whether an underlying multivariate probability distribution differs from a multi-dimensional uniform distribution. Some important properties of the proposed test statistic are discussed. As a special case, the bivariate test statistic is discussed in detail and the critical values of test statistic are obtained. By performing Monte Carlo simulation, the power of the new test is compared with the Distance to Boundary test, which was a recently proposed statistical test for multi-dimensional uniformity by Berrendero, Cuevas and Vazquez-Grande (2006). It has been shown that the test proposed in this thesis is more powerful than the Distance to Boundary test in some cases.