Multivariate statistical analysis has been centered around normal theory thus far. Most papers dealing with this topic are based either directly on the assumption of normality of underlying distributions or indirectly on asymptotic normality of statistics induced by some limiting theorem such as the central limit theorem when sample sizes are sufficiently large. In many if not all fields of application, however, the assumption of normality is not guaranteed even approximately and also the sample sizes are not so large compared with the dimension of the variates in question as to provide a good approximation by normal distributions. Thus, it is required to develop a theory applicable to general multivariate distributions and there are some papers in literature approaching this problem from the standpoint of a distance: [1], [4], [10] and [11] among others. The purpose of this paper is to formulate statistical inference, point and interval estimation as well as testing hypotheses, in terms of a distance or a pseudo distance defined in the family of all probability distributions over a multidimensional Euclidean space. Three specific cases, one of which does not actually give a pseudo distance but which may be useful in some situations, are discussed. The paper consists of four sections; in Section 1 we treat inference with a random sample from one distribution; in Section 2 and Section 3 we treat inference with independent random samples from two and several distributions, respectively, and in the Appendix we present some mathematical results which are used in the preceding sections as tools more practical than [3], [5] and [9] and which may be interesting in themselves.