  We briefly expose some key aspects of the theory and use of dispersion
models, for which Bent Jorgensen played a crucial role as a driving force and
an inspiration source. Starting with the general notion of dispersion models,
built using minimalistic mathematical assumptions, we specialize in two classes
of families of distributions with different statistical flavors: exponential
dispersion and proper dispersion models. The construction of dispersion models
involves the solution of integral equations that are, in general, untractable.
These difficulties disappear when a more mathematical structure is assumed: it
reduces to the calculation of a moment generating function or of a
Riemann-Stieltjes integral for the exponential dispersion and the proper
dispersion models, respectively. A new technique for constructing dispersion
models based on characteristic functions is introduced turning the integral
equations above into a tractable convolution equation and yielding examples of
dispersion models that are neither proper dispersion nor exponential dispersion
models. A corollary is that the cardinality of regular and non-regular
dispersion models are both large.
  Some selected applications are discussed including exponential families
non-linear models (for which generalized linear models are particular cases)
and several models for clustered and dependent data based on a latent Levy
process.
