Multivariate failure time data are commonly encountered in scientific investigations because each study subject may experience multiple events or because there exists clustering of subjects such that failure times within the same cluster are correlated. In this paper, I present a general methodology for analysing such data, which is analogous to that of Liang and Zeger for longitudinal data analysis. This approach formulates the marginal distributions of multivariate failure times with the familiar Cox proportional hazards models while leaving the nature of dependence among related failure times completely unspecified. The baseline hazard functions for the marginal models may be identical or different. Simple estimating equations for the regression parameters are developed which yield consistent and asymptotically normal estimators, and robust variance-covariance estimators are constructed to account for the intra-class correlation. Simulation results demonstrate that the large-sample approximations are adequate for practical use and that ignoring the intra-class correlation could yield rather misleading variance estimators. The proposed methodology has been fully implemented in a simple computer program which also incorporates several alternative approaches. Detailed illustrations with data from four clinical or epidemiologic studies are provided.