  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.
