Likelihood-based statistical tests of competing evolutionary hypotheses (tree topologies) have been available for approximately a decade. By far the most commonly used is the Kishino-Hasegawa test. However, the assumptions that have to be made to ensure the validity of the Kishino-Hasegawa test place important restrictions on its applicability. In particular, it is only valid when the topologies being compared are specified a priori. Unfortunately, this means that the Kishino-Hasegawa test may be severely biased in many cases in which it is now commonly used: for example, in any case in which one of the competing topologies has been selected for testing because it is the maximum likelihood topology for the data set at hand. We review the theory of the Kishino-Hasegawa test and contend that for the majority of popular applications this test should not be used. Previously published results from invalid applications of the Kishino-Hasegawa test should be treated extremely cautiously, and future applications should use appropriate alternative tests instead. We review such alternative tests, both nonparametric and parametric, and give two examples which illustrate the importance of our contentions.