The high luminosity foreseen in the future electron-positron circular collider (FCC-ee) necessitates very intense multi-bunch colliding beams with very small transverse beam sizes at the collision points. This requires emittances comparable to those of the modern synchrotron light sources. At the same time, the stored beam currents should be close to the best values achieved in the last generation of particle factories. This combination of opposite factors represents a big challenge in order to preserve a high beam quality avoiding, at the same time, a machine performance degradation. As a consequence, a careful study of the collective effects and some solutions for the mitigation of foreseen instabilities is required. In this contribution, we discuss the current status of these studies.