Inverse Weibull distribution has been used quite successfully to analyze lifetime data which has non-monotone hazard function. The main aim of this paper is to introduce bivariate inverse Weibull distribution along the same line as the Marshall–Olkin bivariate exponential distribution, so that the marginals have inverse Weibull distributions. The proposed bivariate inverse Weibull distribution has four parameters and it has a singular component. Therefore, it can be used quite successfully if there are ties in the data. The joint probability density function, the joint cumulative distribution function and the joint survival function are all in closed forms. Several properties of this distribution have been discussed. It is observed that the proposed distribution can be obtained from the Marshall–Olkin copula. The maximum likelihood estimators of the unknown parameters cannot be obtained in closed form, and we propose to use EM algorithm to compute the maximum likelihood estimators. We propose to use parametric bootstrap method for construction of confidence intervals of the different parameters. We present some simulation experiments results to show the performances of the EM algorithm and they are quite satisfactory. We provide the Bayesian analysis of the unknown parameters based on very flexible priors. We analyze one bivariate American Football League data set for illustrative purposes, and it is observed that this model provides a slightly better fit than some of the existing models. Finally, we present some generalization to the multivariate case.