AbstractFamilles of asymptotic 100(1 – α)% level confidence bands for the survival function under the general random right‐censorship (GRC) model and the proportional‐hazards model of random right‐censorship, also known as the Koziol‐Green (KG) model, are developed. The family of bands under the GRC model is based on the well‐known product‐limit estimator (PLE), and this family is rich in that it contains as special cases the bands of Hall and Wellner (1980) and Gillespie and Fisher (1979), and more generally, the GF‐type and HW‐type bands of Csörgő and Horváth (1986), as well as new bands not previously studied. The familles of bands under the KG model are based on the maximum‐likelihood estimator ofFunder this particular model. We compare the PLE‐based bands and the MLE‐based bands under the KG model. This enables us to study the loss in efficiency of the former bands when used in a setting where they are not optimal. The notion of asymptotic relative width efficiency (ARWE), defined to be the limiting ratio of the sample sizes needed by the bands to achieve the same asymptotic widths, is employed to compare two bands. Through this efficiency measure it is shown that if the censoring parameter β is known, then the PLE‐based bands are highly inefficient relative to the MLE‐based bands when β is large. When β is not known, the MLE‐based bands are asymptotically conservative. Despite their conservatism, they still dominate the PLE‐based bands when β is not too small or equivalently when the degree of censoring is not too light. We also compare the various PLE‐based bands under the GRC model. The resulting information is valuable for evaluating competing PLE‐based bands. We illustrate the confidence bands by u