This dissertation is concerned with the problem of assessing the fit of a hypothesized parametric family of distributions to data. A nontraditional use of the chi-square and likelihood ratio statistics is considered in which the number of cells is allowed to increase as the sample size increases. A new goodness-of-fit statistic k('2), based on the Pearson correlation coefficient of points of a P-P (percent versus percent) probability plot, is developed for testing departures from the normal, Gumbel, and exponential distributions. A statistic r('2) based on the Pearson correlation coefficient of points on a Q-Q (quantile versus quantile) probability plot is also considered. A new qualitative method based on the P-P probability plot is developed, for assessing the goodness of fit of nonhypothesized probability models to data. This method is not limited to location-scale distributions. Curves were fitted through the Monte Carlo percentiles to obtain formulas for the percentiles of k('2) and r('2) Statistics and Probability; An extensive Monte Carlo power comparison was performed for the normal, Gumbel, and exponential distributions. The statistics examined included those mentioned earlier, statistics based on the moments, statistics based on the empirical distribution function, and the commonly used Shapiro-Wilk statistic. The results of the power study are summarized, and general recommendations are given for the use of these Statistics and Probability;