Systems made by particles in viscoelastic liquids are ubiquitous in a variety of industrial and biological applications. Much work has been done in the last half-century in understanding the effect of non-New-tonian properties on the dynamics of the suspended particles. Theoretical predictions, experimental observations and numerical simulations highlighted peculiar phenomena induced by fluid elasticity that dramatically affect the particle motion and patterning. In this review, the existing literature on the dynamics of non-Brownian particles in viscoelastic fluids is discussed. The main part is focused on the dynamics of rigid particles passively transported in flowing viscoelastic liquids. The available results are classified by increasing level of complexity in terms of hydrodynamic interactions (single-particle problems, binary interactions, multi-body systems) and according to the flow field. Recent results on soft and active viscoelastic suspensions are also discussed. (C) 2014 Elsevier B.V. All rights reserved.