  In this article we survey properties of mixed Poisson distributions and
probabilistic aspects of the Stirling transform: given a non-negative random
variable $X$ with moment sequence $(\mu_s)_{s\in\mathbb{N}}$ we determine a
discrete random variable $Y$, whose moment sequence is given by the Stirling
transform of the sequence $(\mu_s)_{s\in\mathbb{N}}$, and identify the
distribution as a mixed Poisson distribution. We discuss properties of this
family of distributions and present a simple limit theorem based on expansions
of factorial moments instead of power moments. Moreover, we present several
examples of mixed Poisson distributions in the analysis of random discrete
structures, unifying and extending earlier results. We also add several
entirely new results: we analyse triangular urn models, where the initial
configuration or the dimension of the urn is not fixed, but may depend on the
discrete time $n$. We discuss the branching structure of plane recursive trees
and its relation to table sizes in the Chinese restaurant process. Furthermore,
we discuss root isolation procedures in Cayley-trees, a parameter in parking
functions, zero contacts in lattice paths consisting of bridges, and a
parameter related to cyclic points and trees in graphs of random mappings, all
leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed
Poisson distributions naturally arise in the critical composition scheme of
Analytic Combinatorics.
