When the aim of the randomized controlled trial (RCT) is to show that one treatment is superior to another, a statistical test is employed and the trial (test) is called a superiority trial (test). Often a nonsignificant superiority test is wrongly interpreted as proof of no difference between the two treatments. Proving that two treatments are equal in performance is impossible with statistical tools; at most, one can show that they are equivalent. In an equivalence trial, the statistical test aims at showing that two treatments are not too different in characteristics, where "not too different" is defined in a clinical manner. Finally, in a non-inferiority trial, the aim is to show that an experimental treatment is not (much) worse than a standard treatment. In this report, the three types of trials are compared, but the main focus is on the non-inferiority trial. Special attention is paid to the practical implications when setting up a non-inferiority trial. Illustrations are taken from a clinical trial in osteoarthritis and from thrombolytic research.