Completely monotone functions - a digest

This work has a purpose to collect selected facts about the completely monotone (CM) functions that can be found in books and papers devoted to different areas of mathematics. We opted for lesser known ones, and for those which may help determining whether or not a given function is completely monotone. In particular, we emphasize the role of representation of a CM function as the Laplace transform of a measure, and we present and discuss a little known connection with log-convexity. Some of presented methods are illustrated by several examples involving Gamma and related functions.