<p>We present recurrences, combinatorial sums and Bell polynomials for the probability mass function, moments and factorial moments for several compound Poisson distributions, all derived from a common framework. Most workers in the field are motivated to employ a specific (or a few) compound Poisson distribution(s) to fit a particular class of experimental data of interest to the authors. Our focus here is on the properties of the compound Poisson distributions themselves. We treat them using a unified formalism. Many results have been published for individual distributions, sometimes with much effort by the original authors, but we show they are all exemplars of a general formalism. Some new results are presented, as well as fruitful connections between some of the distributions. Several results for the mode and median, which have generally received less attention in the literature, are derived. Some theorems for the mode and median are stated and proved. The results of several numerical studies for the mode and median are presented, such as upper/lower bounds for the values of the mode and the median, also asymptotic expressions for the mode and the median. For some compound Poisson distributions the mode can exceed the median, which never happens for the standard Poisson distribution. An Appendix investigates the scenario where some of the rate coefficients of a compound Poisson distribution are negative. A second Appendix elucidates significant properties of the modal structure of selected compound Poisson distributions. A third Appendix briefly discusses parameter estimation.</p>