The thesis is presented in two parts, (a) "Nonparametric Analysis of Variance", and (b) "An Asymptotic Expansion of the Null Distributions of Kruskal and Wallis's and Friedman's Statistics". In the first part we present a number of new nonparametric tests designed for a variety of experimental situations. These tests are all based on a so-called "matching" principle. The range of situations covered by the tests are: (i) Two-way analysis of variance with a general alternative hypothesis (without interaction). (ii) Two-way analysis of variance with an ordered alternative hypothesis (without interaction). (iii) Interaction in two-way analysis of variance, both the univariate and. multivariate cases. (iv) Latin square designs. (v) Second-order interaction in three-way analysis of variance. (vi) Third-order interaction in four-way analysis of variance. The validity of the tests is supported by a series of simulation studies which were performed with a number of different distributions. In the second part of the thesis we develop an asymptotic expansion for the construction of improved approximations to the null distributions of Kruskal and Wallis's (1952) and Friedman's (1937) statistics. The approximation is founded on the method of steepest descents, a procedure that is better known in Numerical Analysis than in Statistics. In order to implement this approximation it was necessary to derive the third and fourth moments of the Kruskal-Wallis statistic and the fourth moment of Friedman's statistic. Tables of approximate critical values based on this approximation are presented for both statistics.