The authors study the problem of testing whether the distribution function (d.f.) of the observed independent chance variables x_1, \cdots, x_n is a member of a given class. A classical problem is concerned with the case where this class is the class of all normal d.f.'s. For any two d.f.'s F(y) and G(y), let \delta(F, G) = \sup_y | F(y) - G(y) |. Let N(y \mid \mu, \sigma^2) be the normal d.f. with mean \mu and variance \sigma^2. Let G^\ast_n(y) be the empiric d.f. of x_1, \cdots, x_n. The authors consider, inter alia, tests of normality based on \nu_n = \delta(G^\ast_n(y), N(y \mid \bar{x}, s^2)) and on w_n = \int (G^\ast_n(y) - N(y \mid \bar{x}, s^2))^2 d_yN (y \mid \bar{x}, s^2). It is shown that the asymptotic power of these tests is considerably greater than that of the optimum \chi^2 test. The covariance function of a certain Gaussian process Z(t), 0 \leqq t \leqq 1, is found. It is shown that the sample functions of Z(t) are continuous with probability one, and that \underset{n\rightarrow\infty}\lim P\{nw_n < a\} = P\{W < a\}, \text{where} W = \int^1_0 \lbrack Z(t)\rbrack^2 dt. Tables of the distribution of W and of the limiting distribution of \sqrt{n}\nu_n are given. The role of various metrics is discussed.