  This article is an invitation. It is, first, an invitation to consider as a
subject worthy of attention the wide range of situations where small discrete
elements, either bubbles, droplets or solid particles, are embedded in
turbulent flows. Occurring often at a human scale and in our daily
environments, these turbulent dispersed two-phase flows display complex
behavior due to the interplay of two fundamental interactions, the
fluid-particle and particle-particle interactions, compounded by the turbulence
of the carrier flow. This is not a domain where the basic laws are unknown but
where the huge number of degrees of freedom involved call for reduced, or
coarse-grained, statistical descriptions to be developed. Since we are
considering transport and collision phenomena or relaxation processes, it would
seem that they can be handled by kinetic theory. In the general case of
non-fully resolved turbulent flows, we are however dealing with particles
influenced by random media with non-zero time and space correlations. The
second invitation is therefore to recognize the limitations of kinetic-based
descriptions and to address the challenges driving us to extend the classical
framework, for fluid-particle as well as particle-particle interactions. Taking
the standpoint provided by the modern formulation of stochastic processes and
focusing on the description of the particle phase, this review proposes a
step-by-step pedagogical presentation of current models while pointing out new
directions and remaining uncharted territories. This is done to provide answers
to the question `why?' as much as `how?' and to try to kindle interest into
these open and fascinating issues.
