The classification of phase 3 trials as superiority or non-inferiority has become routine, and it is widely accepted that there are important differences between the two types of trial in their design, analysis and interpretation. There is a clear rationale for the superiority/non-inferiority framework in the context of regulatory trials. The focus of our article is non-regulatory trials with a public health objective. First, using two examples from infectious disease research, we show that the classification of superiority or non-inferiority trials is not always straightforward. Second, we show that several arguments for different approaches to the design, analysis and interpretation of superiority and non-inferiority trials are unconvincing when examined in detail. We consider, in particular, the calculation of sample size (and the choice of delta or the non-inferiority margin), intention-to-treat versus per-protocol analyses, and one-sided versus two-sided confidence intervals. We argue that the superiority/non-inferiority framework is not just unnecessary but can have a detrimental effect, being a barrier to clear scientific thought and communication. In particular, it places undue emphasis on tests for significance or non-inferiority at the expense of estimation. We emphasise that these concerns apply to phase 3 non-regulatory trials in general, not just to those where the classification of the trial as superiority or non-inferiority is ambiguous. Guidelines and statistical practice should abandon the sharp division between superiority and non-inferiority phase 3 non-regulatory trials and be more closely aligned to the clinical and public health questions that motivate the trial.