An axiomatic approach is used to develop a one-parameter family of mea- sures of divergence between distributions. These measures can be used to perform goodness-of-fit tests with good statistical properties. Asymptotic theory shows that the test statistics have well-defined limiting distributions which are however analytically intractable. A parametric bootstrap proce- dure is proposed for implementation of the tests. The procedure is shown to work very well in a set of simulation experiments, and to compare favourably with other commonly used goodness-of-fit tests. By varying the parameter of the statistic, one can obtain information on how the distribution that gen- erated a sample diverges from the target family of distributions when the true distribution does not belong to that family. An empirical application analyses a UK income data set.