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This article is an invitation. It is, first, an invitation to consider as asubject worthy of attention the wide range of situations where small discreteelements, either bubbles, droplets or solid particles, are embedded inturbulent flows. Occurring often at a human scale and in our dailyenvironments, these turbulent dispersed two-phase flows display complexbehavior due to the interplay of two fundamental interactions, thefluid-particle and particle-particle interactions, compounded by the turbulenceof the carrier flow. This is not a domain where the basic laws are unknown butwhere the huge number of degrees of freedom involved call for reduced, orcoarse-grained, statistical descriptions to be developed. Since we areconsidering transport and collision phenomena or relaxation processes, it wouldseem that they can be handled by kinetic theory. In the general case ofnon-fully resolved turbulent flows, we are however dealing with particlesinfluenced by random media with non-zero time and space correlations. Thesecond invitation is therefore to recognize the limitations of kinetic-baseddescriptions and to address the challenges driving us to extend the classicalframework, for fluid-particle as well as particle-particle interactions. Takingthe standpoint provided by the modern formulation of stochastic processes andfocusing on the description of the particle phase, this review proposes astep-by-step pedagogical presentation of current models while pointing out newdirections and remaining uncharted territories. This is done to provide answersto the question `why?' as much as `how?' and to try to kindle interest intothese open and fascinating issues.

The dynamics of discrete particles in turbulent flows: open issues and current challenges in statistical modeling