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The question of how particles and droplets can grow in a turbulent environment is of great current interest in many fields, in astrophysics, cloud microphysics, in biology, and in the engineering sciences. For example, coagulation and condensation in turbulent clouds turn microscopic cloud droplets into rain drops. In astrophysics, planetesimals are thought to form by aggregation of microscopic dust grains in the turbulent environment surrounding a forming star. In both cases, turbulence is believed to be a crucial factor for particle growth. Yet the microscopic mechanisms determining this growth are far from understood. In the past few years there has been substantial progress in understanding the mechanisms that determine how particles move in turbulence, albeit mostly for simplified model systems. The challenge is now to understand how these mechanisms lead to rapid particle growth.