Arranged in order of their skewness, here are some models which produce overdispersed counts.

• Thomas double Poisson:
  arises from a Poisson distributed population, the reproductive success of which is Poisson distributed, and the progeny are subject to random but limited dispersal.
• Neyman contagious type A:
  arises from a Poisson distributed population, each member of which gives rise to equal-sized groups randomly distributed around the original members.
• Polya-Aeppli:
  randomly distributed colonies where the number of individuals per colony is geometrically distributed.
• Negative binomial:
  among many other mechanisms - a continuously reinvading Thomas double Poisson. Or Poisson distributed samples, with mean λ, which is gamma-distributed.
• Poisson lognormal (or discrete lognormal):
  Samples are Poisson distributed, with mean λ, which is lognormally-distributed.