The observation of randomness patterns serves as guidance for the craft of probabilistic modelling. The most used count models—Binomial, Poisson, Negative Binomial—are the discrete Morris’ natural exponential families whose variance is at most quadratic on the mean, and members as well of the Katz-Panjer, Power Series and Generalized Hypergeometric families, which accounts for their many advantageous properties. Some other basic count models are also described, as well as models with less obvious but useful randomness patterns in connection with maximum entropy characterisations, such as Zipf and Good models. Simple tools, as truncation, thinning, or parameters randomisation, are straightforward ways of constructing other count models. Some of them are useful for understanding biological phenomena, such as modelling the number of extra-pair nestlings in broods.