Many biological and medical studies have as a response of interest the time to occurrence of some event, X, such as the occurrence of cessation of smoking, conception, a particular symptom or disease, remission, relapse, death due to some specific disease, or simply death. Often it is impossible to measure X due to the occurrence of some other competing event, usually termed a competing risk. This competing event may be the withdrawal of the subject from the study (for whatever reason), death from some cause other than the one of interest, or any eventuality that precludes the main event of interest from occurring. Usually the assumption is made that all such censoring times and lifetimes are independent. In this case one uses either the Kaplan-Meier estimator or the Nelson-Aalen estimator to estimate the survival function. However, if the competing risk or censoring times are not independent of X, then there is no generally acceptable way to estimate the survival function. There has been considerable work devoted to this problem of dependent competing risks scattered throughout the statistical literature in the past several years and this paper presents a survey of such work.