Categorical variables are of uttermost importance in biomedical research. When two of them are considered, it is often the case that one wants to test whether or not they are statistically dependent. We show weaknesses of classical methods -- such as Pearson's and the G-test -- and we propose testing strategies based on distances that lack those drawbacks. We first develop this theory for classical two-dimensional contingency tables, within the context of distance covariance, an association measure that characterizes general statistical independence of two variables. We then apply the same fundamental ideas to one-dimensional tables, namely to the testing for goodness of fit to a discrete distribution, for which we resort to an analogous statistic called energy distance. We prove that our methodology has desirable theoretical properties, and we show that we can calibrate the null distribution of our test statistics without resampling. We illustrate all this in simulations, as well as with some real data examples, demonstrating the adequate performance of our approach for biostatistical practice.