Suspensions with deformable solid inclusions have both scientific and industrial interest. Examples include microgel particle suspensions, biological fluids, and filled polymers. Due to the interplay of numerous factors (e.g., particle size, shape, and mechanical properties, suspending liquid rheology and flow conditions, flow cell shape, interactions between the particles and the flow channel and among the particles), these systems can show many different flow behaviors. The study of the flow behavior of soft particle suspensions is an open field and is attracting an increasing interest from the scientific community. As elastic beads are good models for biological cells, one reason for such interest is related to the possibility of exploiting particle mechanical-property-dependent flow behavior to diagnose diseases. On the other hand, in the recent years, suspensions of elastic microparticles are also proving useful in a completely different technological field, namely oil recovery from porous media. This paper gives an overview of the state of the art and envisages possible future perspectives on the dynamics, rheology, and applications of elastic particle suspensions.