Some recent statistical results for infinitely divisible distributions are presented. The class of infinitely divisible distributions is shown to provide useful formulations for problems involving heavy-tailed distributions and for problems involving convolutions. Secondly, a generalized, parametric theory of multivariate statistical analysis based on infinitely divisible distributions is outlined. Because the infinitely divisible class includes the normal family, this theory is more general than that based on multivariate normal distributions. The corresponding methods are tractable because they are based on parameters, analogous to covariances, which are easy to estimate and interpret. Thirdly, a method of inference for infinitely divisible time series is outlined.