While violation of the homogeneity of variance assumption has received considerable attention, violation of the assumption of normally distributed data has not received as much attention. As a result, researchers may have the mistaken impression that as long as the assumptions of independence of observations and homogeneity of variance are satisfied, violations of the distributional assumption gave inconsequential effects. This paper reviews some of the relevant literature and reports the results of a new Monte Carlo study indicating that this is not the case. The simulation investigated the effects of varying skewness and kurtosis levels, while maintaining equal population variances, on the two-sample "t" test and Welch's robust "t" test. Sample sizes were either small or moderate, and equal or unequal. Results indicate that, with skewed distributions, the validity of both the "t" test and the Welch test clearly depends on the two distributions being skewed in the same direction. When the two parent distributions are skewed in the same direction, both tests have quite acceptable Type I error rates, even with relatively small samples. However, when the two parent distributions are skewed in opposite directions, then the true Type I error rates can deviate markedly from the nominal level even though population variances are equal. The actual Type I error rate of the "t" test performed at a 0.05 nominal level with homogeneous variances can be higher than 0.08 with a two-tailed test, and can be higher than 0.11 with a one-tailed test. (Contains 16 figures and 23 references.) (SLD)