Summary. The paper is motivated by cure detection among the prostate cancer patients in the National Institutes of Health surveillance epidemiology and end results programme, wherein the main end point (e.g. deaths from prostate cancer) and the censoring causes (e.g. deaths from heart diseases) may be dependent. Although many researchers have studied the mixture survival model to analyse survival data with non-negligible cure fractions, none has studied the mixture cure model in the presence of dependent censoring. To account for such dependence, we propose a more general cure model that allows for dependent censoring. We derive the cure models from the perspective of competing risks and model the dependence between the censoring time and the survival time by using a class of Archimedean copula models. Within this framework, we consider the parameter estimation, the cure detection and the two-sample comparison of latency distributions in the presence of dependent censoring when a proportion of patients is deemed cured. Large sample results by using martingale theory are obtained. We examine the finite sample performance of the proposed methods via simulation and apply them to analyse the surveillance epidemiology and end results prostate cancer data.