Multivariate survival data typically have correlated failure times. The correlation is often the consequence of the observational design, e.g. clustered sampling, matching or repeated measures. We assume that the correlation between the failure or survival times can be accounted for by random effects or frailties in the hazard. We focus attention here on substantive problems where the random effects are not a nuisance, but are of primary interest as they have an explanatory role, for example, in genetic studies and longitudinal studies of recurrent or multiple events in which the processes operating at the individual level are under investigation. We present various analytically tractable random effects models for multivariate survival data. The paper contains two illustrative examples. The first concerns a treatment trial of heart patients and examines the times to onset of chest pain brought on by three endurance exercise tests. The second examines social and genetic effects in the association of ages to first marriage of twins.