Cumulants of infinitely divisible distributions

In this expository article, we provide a unified version of the literature on certain aspects of cumulants of infinitely divisible distributions. In view of the spectral representations corresponding to the distributions referred to, with appropriate modifications to standard arguments, it follows that the cumulants of these distributions possess some nice features. Giving several illustrative examples, we attempt to explain as to what these mean. Although the emphasis of the paper is on univariate distributions, we briefly touch upon the multivariate case to verify whether this latter case has something new to offer to us on the subject.